PARTIAL DIFFERENTIAL EQUATIONS FROM MATRICES WITH ORTHOGONAL COLUMNS

. We discuss a system of third order PDEs for strictly convex smooth functions on domains of Euclidean space. We argue that it may be understood as a closure of sorts of the ﬁrst order prolongation of a family of second order PDEs. We describe explicitly its real analytic solutions and all the solutions which satisfy a genericity condition; we also describe a family of non-generic solutions which has an application to Poisson geometry and Kahler structures on toric varieties. Our methods are geometric: we use the theory of Hessian metrics and symmetric spaces to link the analysis of the system of PDEs with properties of the manifold of matrices with orthogonal columns.


Introduction
Let φ be a strictly convex smooth function defined on a connected subset Ω ⊂ R n .Its Hessian Hφ defines at each point an inner product and therefore the inverse of the Hessian matrix is also a smooth field of inner products.It is natural to ask whether this field is also the Hessian of a function.If g is a field of inner products which is the Hessian of a function, then there must be an equality of partial derivatives: If Ω has trivial first homology group, then the agreement of the above partial derivatives implies that g is the Hessian of a function [3].We shall assume from now on that the domain Ω as trivial first homology group.
Definition 1.0.1.A strictly convex function φ ∈ C ∞ (Ω) has property I if it satisfies the system of third order PDEs: The purpose of this paper is to analyze the system of third order PDEs (1) for strictly convex functions.
To be more precise about our focus, we note that it is possible to construct strictly convex solutions to (1) by elementary means: Every strictly convex function of one variable has property I.If φ 1 (x 1 ) and φ 2 (x 2 ) are strictly convex functions of one variable, then φ 1 (x 1 ) + φ 2 (x 2 ) has property I.Such a function solves the second order hyperbolic PDE with constant coefficients and, conversely, all the solutions of ( 2) decompose (locally) as the sum of two functions on each of the variables x 1 and x 2 ; the parallel translates of the coordinate axis are the (constant) characteristics of the solutions.If (2) is replaced by any second order hyperbolic PDE with constant coefficients whose solutions have (constant) orthogonal characteristics, then its strictly convex solutions will have property I.
There is a natural generalization of this family of hyperbolic second order PDEs to arbitrary dimensions.Its strictly convex solutions, which we refer to as functions with (constant) orthogonal characteristics, will also have property I.
It is thus natural to study 'how close' a strictly convex function with property I may be from having orthogonal characteristics.
Our main results describe sufficient conditions for a strictly convex function with property I to have orthogonal characteristics.Among such sufficient conditions there is a generic one: Theorem 1.0.2.If a strictly convex function has property I and at every point the eigenvalues of its Hessian are simple, then it has orthogonal characteristics.
Another sufficient condition refers to the regularity of the functions: Theorem 1.0.3.If a real analytic strictly convex function has property I, then it has orthogonal characteristics.
We shall prove Theorem 1.0.2first for functions of two variables.For them the system (1) has two equations and the theorem will follow from an algebraic manipulation valid under the hypothesis on the Hessian.The algebraic manipulation will have a geometric counterpart: The family of second order hyperbolic PDEs with constant coefficients whose solutions have orthogonal characteristics defines a pencil of hyperplanes on the space of jets of order two; away from its base, and in the set where the Hessian is strictly positive, it restricts to a foliation.Strictly convex functions with property I define a subset 1 of the space of jets of order three.The hypothesis on the eigenvalues of the Hessian singles out the locus of smooth points for which the jet projection is a submersion; its image is the aforementioned foliated open subset of the jets of order two.The geometric manifestation of our algebraic manipulation will be that the Cartan connection is tangent to the leaves of the pullback foliation.This is why one may say that for strictly convex functions the system of third order PDEs (1) is the closure of the prolongation of the aforementioned pencil of second order hyperbolic PDEs.
To go to arbitrary dimensions we will not follow the jet space approach, as we find the algebraic complexities difficult to manage.We shall switch our viewpoint to that of Hessian metrics.In this language what we are asking is when for a given Hessian metric on a domain of Euclidean space its inverse metric is also Hessian.We refer to such metrics as Hessian metrics with property I.The first manifestation of the relevance of the metric viewpoint will be the following: with connection [3], π : (Gl(n) + , ∇) → P, where P denotes positive matrices and π and ∇ are a natural map and connection, respectively, which come from symmetric space theory.We shall argue that the universal orthogonal frame bundle offers an appropriate replacement for the jet space picture.Briefly, jet spaces of order two will be replaced by positive matrices P; the subset of the jet spaces of order three defined by property I will be replaced by the submanifold of matrices with orthogonal columns C ⊂ Gl(n) + ; the restriction of the jet projection will correspond to π| C : C → P; the Cartan connection will correspond to the universal Levi-Civita connection ∇.Property I for a Hessian metric will translate as follows: Theorem 1.0.5.A Hessian metric Hφ in Ω has property I if and only if for any point x ∈ Ω and any curve γ at x ∈ Ω there exist an orthonormal frame for Hφ(x) in C such that the corresponding horizontal lift of γ at that frame is tangent to C.
There will be a property analogous to the tangency of the Cartan connection to the pullback foliation coming from the pencil of degree two hyperbolic PDEs: The generic condition on eigenvalues in Theorem 1.0.2 is just the open stratum of a natural stratification of P. To describe more precise sufficient conditions for a Hessian metric with property I to come from a function with orthogonal characteristics, we will analyze the interaction among this stratification, the foliation on C defined by the universal Levi-Civita connection, and the map π| C .Theorem 1.0.3 will hinge on real analytic features of these objects.
As we shall see, property I for strictly convex functions appears in a problem of Poisson geometry in toric varieties.The so-called totally real toric Poisson structures have properties analogous to that of Hamiltonian Kahler forms.For instance, whereas the latter are encoded by appropriate strictly convex functions [6], the former are encoded by the simplest strictly convex functions: quadratic forms.The most natural Poisson-theoretic PDE for a pair given by a totally real toric Poisson structure and a Hamiltonian Kahler form will correspond to property I: Theorem 1.0.7.Let (X, T) be a (smooth) toric variety endowed with a totally real toric Poisson structure Π and a Kahler form σ for with the action of the maximal compact torus T ⊂ T is Hamiltonian.Let P denote the inverse Poisson structure to σ.Then the following statements are equivalent: (1) The Poisson structures Π and P Poisson commute: [Π, P ] = 0.
(2) In a basis of the Lie algebra of T for which Π corresponds to the standard quadratic from of R n , the strictly convex function which corresponds to σ has property I.
In complex dimension one a totally real toric Poisson structure and (the inverse of) a Hamiltonian Kahler form always Poisson commute because the commutator is a field of trivectors on a surface; equivalently, if we use Theorem 1.0.7 this corresponds to the fact that all strictly convex functions of one variable have property I.As it will turn out Theorem 1.0.3 will imply that in the real analytic category such a commuting pair is the Cartesian product of one dimensional commuting pairs : Theorem 1.0.8.Let (X, T) be a projective toric Poisson variety endowed with a totally real toric Poisson structure Π which Poisson commutes with a real analytic Hamiltonian Kahler structure σ.Then (X, T) is a Cartesian product of projective lines and both Π and σ factorize.
We shall also describe a family of strictly convex functions which satisfy property I but which do not have orthogonal characteristics.We will use it to construct Hamiltonian Kahler forms in certain (T -invariant) regions of toric varieties.These regions can be thought of as the result of gluing to a (T -round) 0-handle several (T -round) 1-handles.An illustration of the construction is the following: Proposition 1.0.9.Let U ⊂ CP 2 be the complement of small T 2 -invariant neighborhoods of [1 : 0 : 0] and [1 : 0 : 1].Then there exist a totally real toric Poisson structure on CP 2 and a Hamiltonian Kahler form on U which Poisson commute.
The structure of this paper is as follows.Section 2 describes how matrices with orthogonal columns are used to define the family of differential relations of second order with constant coefficients whose solutions we call functions with (constant) orthogonal characteristics; we also discuss why they have property I.In Section 3 we do the analysis of the system of third order PDEs (1) for strictly convex functions of two variables using jet spaces.The viewpoint of Hessian metrics is introduced in Section 4. Property I is translated as symmetry of the Christoffel symbols, an algebraically simpler condition which allows to analyze the interaction of property I with the Legendre transform.Section 5 describes how the universal orthogonal frame bundle offers the appropriate setting for the geometric analysts of property I.We analyze the map π : (C, ∇) → P from the submanifold of orthogonal matrices with the restriction of the universal Levi-Civita connection onto the manifold of positive matrices; this is the our replacement of the subsets defined by property I in the space of jets of order three and two with the restriction of the Cartan connection.Section 6 contains our main results: Firstly, the description of property I as a differential relation related to the submanifold of orthogonal matrices C ⊂ (Gl(n) + , ∇) → P. Secondly, sufficient conditions for a Hessian metric with property I to come from a strictly convex function with orthogonal characteristics.In Section 7 we describe a family of strictly convex functions which have property I but do not have in general orthogonal characteristics.The domains of definition of its members are what we call polytopes with 1-handles.Despite polytopes with 1handles not being convex in general, we show that the family is invariant under Legendre transform.Section 8 contains our applications to Poisson geometry.We explain how on a smooth toric variety the Poisson commuting equation for a totally real toric Poisson structure and for (the inverse of) a Hamiltonian Kahler form can be rewritten as property I for either the Kahler or the symplectic potential [6] of the latter form.That allows us to conclude that in the real analytic category any such commuting pair must be the Cartesian product of commuting pairs on projective lines.We also use the family introduced in Section 7 to construct commuting pairs on certain topologically non-trivial regions of toric varieties (which are not Cartesian products).
The author is grateful to Max Planck Institute for Mathematics in Bonn for its hospitality and finantial support.

Solutions with orthogonal characteristics
It is possible to construct strictly convex functions with property I by elementary means.and, therefore, if Hφ(x) −1 is the Hessian of ψ(x), then (Hφ(Bx)) −1 is the Hessian of ψ(Bx).
A function is (locally) of the form φ = φ 1 (x 1 ) + • • • + φ n (x n ) if and only if it is a solution of the system of second order PDEs The solutions of the system have (constant) characteristics given by the collection of axis.This information can be used to rewrite (3) in a more geometric fashion.For any n × n matrix A one can define a differential operator of order two on functions with values on matrix-valued functions by the following recipe: Equivalently, the ij-th component is the Lie derivative of φ with respect to the (constant) vector field defined by the i-th column of A, followed by the Lie derivative with respect to the vector field defined by the j-th column.Let D denote the set of diagonal matrices with strictly positive entries and let D(Ω) denote smooth functions on Ω with values on D. It follows that a function φ is strictly convex and satisfies (3) if and only if L 2 I φ ∈ D(Ω), where I is the identity matrix.Let C denote the set of matrices with orthogonal columns.
Because D is invariant by conjugation by permutation matrices, in Definition 2.0.1 we may assume that C has positive determinant.We will abuse notation and use C to refer to matrices with orthogonal columns and positive determinant.C φ = C T HφC = ΛB T HφBΛ and (5) we deduce that B T HφB ∈ D(Ω), or, equivalently, that Hφ(Bx) ∈ D(Ω).Thus, locally and φ i is strictly convex.Therefore φ(Bx) is strictly convex and has property I, and so the same occurs for φ(x) = φ(B −1 (Bx)).

The two-dimensional case
We would like to know whether there exist strictly convex functions with property I which do not have orthogonal characteristics.For that we find convenient to discuss the algebraic structure of the system of PDEs (1).This should be easier in the lowest non-trivial dimension.
We shall denote partial derivatives of a function φ(x), x = (x 1 , x 2 ) ∈ Ω ⊂ R 2 , by means of subindices which follow a comma.We introduce independent variables to parametrize (homogeneous) jet spaces of order two and three: Strictly convex functions correspond to the open subset χζ − τ 2 > 0, χ + ζ > 0. The system of PDEs (1) corresponds to the common solutions of the degree three homogeneous polynomials: Lemma 3.0.1.Strictly convex functions with property I correspond in the space of jets of order three of functions in the plane to an open subset of an intersection of quadrics: Proof.We interpret the equations of the cubics (6) as a (non-homogeneous) linear system with indeterminates νζ + χξ − 2τ ω and υζ + χω − 2τ ν: In the open subset defined by χζ − τ 2 = 0 we obtain the equivalent relations To each [a : b] ∈ RP 1 one can associate the following second order PDE with constant coefficients for strictly convex functions: It corresponds to open subset of a hyperplane of the space of jets of order two Proposition 3.0.2.Let φ ∈ C ∞ (Ω) be a strictly convex function.
(2) If φ satisfies (8) for more than one [a : b] ∈ RP 1 , then φ is -up to a degree one polynomial -a multiple of the standard quadratic form x 2 1 + x 2 2 .(3) If φ has property I and its Hessian has simple eigenvalues, then φ satisfies (8) for some [a : b] ∈ RP 1 .
Proof.The set of equations (8) are exactly those second order PDEs whose solutions have orthogonal characteristics.Therefore item (1) is the specialization of Lemma 2.0.2 to the two-dimensional case.Alternatively, item (1) follows from the inclusion of the solutions of (10) in the solutions of (7).
If φ satisfies (8) for more than one [a : b] ∈ RP 1 , then its second jet belongs to the base of the pencil (9): χ − ζ = τ = 0.That is to say φ ,12 = 0 and φ ,11 = φ ,22 .Therefore 0 = φ ,221 = φ ,111 = φ ,112 = φ ,222 .Hence φ is a degree two polynomial whose homogeneous part of degree two equals k(x 2 1 + x 2 2 ), k > 0. The Hessian Hφ has two eigenvalues if and only if it misses the base of the pencil.Equivalently, the field of vectors in the plane (φ ,22 − φ 11 , φ ,12 ) ∈ R 2 has no zeroes.Therefore we can (locally) take the quotient of the components of the vector field to get a well-defined slope function.Property I as in (7) can be rewritten where •, • is the standard inner product.This implies that the slope function is constant, which is exactly the second order PDE (8) for some [a : b] ∈ RP 1 .Proposition 3.0.2does not clarify whether strictly convex functions with property I and which do not have orthogonal characteristics exist.As we shall discuss in Section 7 such solutions exist: It is possible to start from a multiple of the standard quadratic form in a subdomain of Ω which 'bifurcates' into solutions to different equations in (8) in other subsets of the domain Ω.
Remark 3.0.3.(The Cartan connection on jet spaces) The algebraic manipulation in item (3) in Proposition 3.0.2has a geometric counter-part.The requirement on the Hessian corresponds to the regularity condition needed to identify solutions with holonomic sections with respect to the Cartan connection: On the one hand, the subset of the jet spaces which corresponds to property I is not smooth; the 1-forms On the other hand, the smooth locus of the intersection of quadrics fails to be transverse to the fibers of the projection onto jets of order two in the points over the base of the pencil.
The connection 1-forms one has to add when passing from jets of order two to jets of order three are: The pullback foliation is defined by the 1-forms The equalities hold in the intersection of (7) with the complement of the pullback of the base of the pencil.Therefore holonomic sections in this subset are tangent to the pullback foliation.Hence their order two jet must be inside a hyperplane of the pencil.
Remark 3.0.4.(Orthogonal characteristics and Legendre transform) Let φ be an strictly convex function on a convex domain Ω ⊂ R 2 .Its Legendre transform is an strictly convex function φ * on a convex domain Ω * which is related to Ω by a (Legendre) diffeomorphism.Let us assume that φ satisfies the constant coefficients second order PDE: Because Hφ * at x ∈ Ω * equals Hφ −1 at its related point in Ω we have the equality aφ * ,22 + cφ * ,11 + bφ * ,12 = 0. Thus the Legendre transform induces an involution in the parameter space of constant coefficients degree two homogeneous PDEs: [a : To the point [1 : 1 : 0] corresponds the Laplace equation, which has no strictly convex solutions.The projective line [a : −a : b] parametrizes hyperbolic PDEs with orthogonal characteristics (8).Therefore if φ is a function on the convex domain Ω with orthogonal characteristics, so its Legendre transform is.This invariance property holds regardless of the dimension: , where the first equivalence uses the relation between Hessian matrices of the original function and its Legendre transform, and in the second equivalence we have inverted the matrices and we have used C T C ∈ D.

Hessian metrics with symmetric Christoffel symbols
To generalize the results in Section 3 the complexities brought by the increase of dimension shall be dealt with by shifting the perspective to that of Hessian metrics.
A Hessian metric on Ω ⊂ R n is a Riemannian metric obtained as the Hessian matrix of a (strictly convex) function on Ω. Property I for strictly convex functions can be translated to a requirement for a Hessian metric: that its inverse metric be Hessian as well.In such case we say that the given Hessian metric has property I.
There is another natural differential condition on Hessian metrics which allows to formulate in arbitrary dimensions the algebraic simplification of property I described in Lemma 3.0.1.For a Hessian metric Hφ the Christoffel symbols of the first kind equal the partial derivatives of order three: Γ ijk = φ ,ijk .The Christoffel symbols (of the second kind) are ,k denote the partial derivative with respect to k of the entries of the Hessian matrix.For each 1 ≤ k ≤ n we define the Christoffel matrix Property I corresponds to an open subset of the solutions of a system of polynomial equations of degree 2n − 1 in the space of jets of order three.The symmetry of the Christoffel symbols is determined by a system of polynomial equations of degree n; for n = 2 it is exactly (7).The generalization of Lemma 3.0.1 to arbitrary dimensions is that property I translates into the symmetry of Christoffel symbols: Proof of Lemma 1.0.4.The Hessian metric is invertible if and only if the i-th and j-th lines of [Hφ −1 ] ,j and [Hφ −1 ] ,i are equal.This is equivalent to the same condition for the matrices [Hφ −1 ] ,j Hφ and [Hφ −1 ] ,i Hφ.If we prolong the identity Hφ −1 Hφ = I, then the condition transforms on the same condition for the Christoffel matrices Γ j and Γ i .This amounts to symmetry of all Christoffel matrices.
The problem of the symmetry of Christoffel symbols of Hessian metrics is amenable to Lie theoretic methods.A first instance of that is the following: Proof.Let s be the vector subspace of symmetric matrices and let d ⊂ s denote the diagonal matrices; this is a maximal Cartan subalgebra of the symmetric matrices.The Christoffel matrix Γ k is the product of the symmetric matrices Hφ −1 and [Hφ] ,k .Therefore Hφ has symmetric Christoffel matrices if and only if the following commutators are trivial: This is equivalent to require that [Hφ] ,k be in the same maximal torus of s as Hφ −1 .If B is a orthogonal matrix which diagonalizes Hφ then it also diagonallizes Hφ −1 : Therefore if Hφ, [Hφ] ,k are in the Cartan subalgebra Ad B (d) ⊂ s, then so is Hφ −1 .This shows the equivalence between (1) and (2).
If (2) holds then remains in the same Cartan subalgebra of the commuting factors, which proves (3).Condition (3) by definition implies the symmetry of the Christoffel matrices.
By Lemma 1.0.4Hessian metrics with symmetric Christoffel symbols are the same as Hessian metrics with property I. Thus by Lemma 2.0.2 strictly convex functions with orthogonal characteristics define Hessian metrics with symmetric Christoffel symbols.We can reprove this result with a Lie theoretic approach: for some C ∈ C, then the Christoffel matrices of Hφ for all points in Ω are in the Cartan subalgebra Ad C (d).
In particular Hφ has symmetric Christoffel symbols.
Proof.By hypotheses for each x ∈ Ω Hence Hφ = (C T ) −1 ΛC −1 and upon taking its first order prolongation Therefore If we push forward each equation in (11 by the Legendre diffeomorphism dφ, then the Lie derivative of the pushed forward functions -entries of the inverse Hessian -by the pushed forward vector fields will subtract to zero as well.The entries of the inverse Hessian matrix are pushed forward to the entries of the Hessian of φ * ; the coordinate vector fields are pushed forward to the columns vector fields of the Jacobian matrix Hφ, which at points in Ω * is the matrix Hφ * −1 .Therefore property I for Hφ is equivalent to which is the symmetry of the Christoffel matrices of Hφ * .Therefore by Lemma 1.0.4Hφ * has property I.

The universal frame bundle for Hessian metrics and matrices with orthogonal columns
To generalize Proposition 3.0.2 to arbitrary dimensions jet spaces will be replaced by (a subset of) the principal orthogonal frame bundle of the Hessian metric with its Levi-Civita connection.There are three reasons to do that: Hφ splits (locally, but along the same characteristics everywhere).In other words, the conclusion of the de Rham Splitting Theorem holds.Therefore to study the relation between property I and orthogonal characteristics it may be appropriate to look at parallel transport on the principal frame bundle with its the Levi-Civita connection.
(c) Hessian metric on domains of Euclidean space are characterized among Riemannian metrics as those whose frame bundle is the pullback of a universal principal bundle with connection coming from symmetric space theory [3].For a function φ the information of the homogeneous part of its second jet is the same as the one contained in its Hessian.Thus for our strictly convex functions we shall be looking at the map x → Hφ(x), which takes values in the positive matrices P. There, the pencil in (9) defined by hyperbolic PDEs with orthogonal characteristics generalizes as follows: The second order PDE equation (3) corresponds to Hessian metrics with image in the (positive) diagonal matrices D. Matrices with orthogonal columns have a factorization into an orthogonal and a diagonal matrix.Thus we may confine ourselves to the family of second order PDEs To each of them there corresponds the subset Ad B (D) ⊂ P; their union over B ∈ SO(n) fills P as any positive matrix can be diagonalised by a special orthogonal transformation.
For a Riemannian metric defined on a subset of Euclidean space, its orthogonal frame bundle -forgetting for the moment about the Levi-Civita connectionis constructed via pullback: The map π : Gl(n has image the closed embedded submanifold of positive matrices.The restriction to its image π : Gl(n) + → P (12) • is a (right) principal bundle for SO(n); • intertwines the right action of SO(n) on Gl(n) + and the adjoint action of SO(n) on P; • is the bundle of (positively oriented) orthogonal frames for metrics on R n .Let ∇ be the SO(n)-invariant principal connection on π : Gl(n) + → P which at the identity matrix has as horizontal space the symmetric matrices2 .Proposition 5.0.1.[3, Proposition 4.1] If Hφ is a Hessian metric on Ω, then the pullback of ∇ by Hφ : Ω → P, x → Hφ(x), is the Levi-Civita connection on the orthogonal frame bundle of Hφ.Furthermore, this property characterizes Hessian metrics among Riemmanian metrics in domains of Euclidean space.
The appropriate replacement of the jets of order three will not be the full bundle of orthogonal frames.It will be the subset of matrices with orthogonal columns.The following result, of which Proposition 1.0.6 in the Introduction follows, shows that it is well-behaved with respect to the universal Levi-Civita connection: Proof.Let ι be the inversion map on Gl(n) + and q = π•ι : Gl(n) + → P, A → A T A.
A matrix C has orthogonal columns if and only if q(C) ∈ D. Therefore C is the preimage under a submersion of the closed embedded submanifold of positive diagonal matrices, thus a closed embedded submanifold of Gl(n) + .We have already used the (unique) factorisation of a matrix with orthogonal columns as a product of an orthogonal and a diagonal matrix.It is straightforward that it gives rise to a Cartesian product of manifolds C = SO(n)D.The product structure in (1) implies that its tangent space at The horizontal space of ∇ there is C • s.Because the conjugation of a skew orthogonal matrix by an orthogonal one can never be symmetric, the intersection of the tangent spaces must be C • d.Therefore the intersection of the horizontal space of ∇ with T C is the distribution 3 tangent to the left translates of D by SO(n).
Next, we argue how π : (C, F ) → P provides a 'desingularization' of the pencil Ad B (D), B ∈ SO(n).
Proposition 5.0.3.The restriction π| C : C → P has the following properties: (1) It is a surjective map all whose values are clean.
(2) The restriction of the differential of π| C to T F has trivial kernel and the restriction of π| C to the leaf BD is a diffeomorphism onto Ad B (D).
Proof.Let V ∈ P. Then it diagonalizes in an orthogonal basis: where the latter subgroup is the stabilizer of Λ for the adjoint action.The kernel of the differential of π at BΛ 1/2 is BΛ 1/2 • so(n).The tangent space of C at BΛ 1/2 is BΛ 1/2 • ad −1 Λ (d).Because the adjoint orbit through Λ intersects D cleanly at Λ, their intersection -which is the kernel of of the differential of π| C at BΛ 1/2 -is BΛ 1/2 • so(n) Λ .Therefore all values of π| C are clean.
The tangent space to the leaf of F through BΛ 1/2 is BΛ 1/2 • d.Its intersection with BΛ 1/2 • so(n) Λ is trivial.Therefore the restriction of π to BD is a local diffeomorphism over its image.That image is by construction Ad B (D).To conclude that it is a diffeomorphism one can either check that the map is bijective or argue that the manifolds involved are contractible.
The base of the pencil (9) corresponds to inner products in the plane which have a unique eigenvalue.In arbitrary dimensions we have an analogous subsets.For each symmetric matrix we can order its eigenvalues (with their multiplicity) in an increasing sequence.To each partition κ of {1, . . ., n} there correspond a subset Θ κ s ; likewise, to each matrix with orthogonal columns we can order the norm of its columns in an increasing sequence.In that way we obtain partitions of D, P and C: Θ D , Θ P , Θ C .Proposition 5.0.4.The partitions Θ D , Θ P and Θ C are stratifications of D, P and C, respectively, and they interact with the map π| C : (C, F ) → P as follows: (1) The preimage of Θ κ P is Θ κ C and the restriction is a principal bundle: 3 One could also deduce involutivity by recalling that the curvature of the connection is and, therefore, the abelian subalgebra d is flat.
(2) The foliation F = SO(n)D of C intersects the stratum Θ κ C cleanly and induces there the foliation SO It is in this sense that π : (C, F , Θ C ) → (P, Ad SO(n) (D), Θ P ) is a desingularization of the stratified pencil.
Proof.The group SO(n) acts on s by conjugation.As for any proper action it produces a stratification of s in orbit types [4, Chapter 2]: two symmetric matrices are related if their isotropy subgroups are conjugated.It is well known that upon passing to connected components the outcome is a (Whitney B) stratification of s.The stratification Θ s is the result of possibly collecting some of the strata of the orbit type stratification belonging to the same subset of the orbit type partition; in any case, it is still a stratification for the partial order associated to the partitions κ of {1, . . ., n}.The stratification Θ s -made of adjoint orbits -intersects the Cartan subalgebra d cleanly, thus inducing a stratification Θ d .Each strata there is a face of the positive Weyl chamber of diagonal matrices with diagonal elements ordered increasingly; an open convex polytope in a vector subspace d κ .The partition Θ P is obtained by intersecting Θ s with the open subset of positive matrices, thus it is a stratification.The partition Θ D is also obtained upon intersection; it is a stratification because for instance D is an open subset of d.Finally, Θ C is the pull back of Θ D by the submersion q, and therefore it is a stratification as well.
Let C ∈ C with factorisation C = BΛ.Then q(C) = BΛ −2 B T and therefore k is an open subset of the vector subspace d κ of all matrices whose stabilizer contains SO(n) κ .Because the fiber of π| C through C is CSO(n) κ and π| Θ κ C is saturated by fibers of π| C , it is a principal SO(n) κ -bundle.This proves (1).The fibers of q are the orbits of the left SO(n)-action.The restriction q| D : D → D is the square map, which preserves the strata of Θ D .Therefore the factorization of C is compatible with the stratification:

Differential relations on the submanifold of matrices with orthogonal columns
We want to transfer property I for Hessian metrics into a differential condition for the orthogonal frame bundle at the submanifold of matrices with orthogonal columns.
Upon choosing an orthonormal frame for Hφ(x), we can construct the horizontal lift of Hφ(γ) based at the orthonormal frame.Definition 6.0.1.A Hessian metric Hφ in Ω has property C if for any point x ∈ Ω and any curve γ at x ∈ Ω there exist an orthonormal frame C ∈ C for Hφ(x) such that the corresponding horizontal curve is tangent to C at C.

We now translate property I to the universal orthogonal frame bundle setting:
Proof of Theorem 1.0.5.We must show that a Hessian metric in Ω ⊂ R n has property C if and only if it has symmetric Christoffel symbols.
Property C is linear in the velocity of the curve at x. Thus it is enough to prove the equivalence for γ(t) = x + te k , 1 ≤ k ≤ n.Let us denote the horizontal lift at A ∈ C by A(t).That A belongs to C means that A T A = Λ ∈ D. By Proposition 5.0.1 (taken from [3]) the pullback of π : (Gl(n) + , ∇) → P by Hφ is the orthonormal frame bundle of Hφ with its Levi-Civita connection.Thus we have: The image by the differential of q of the vector of Therefore the Hessian metric satisfies property C at A if and only if We have Γ k = Hφ −1 [Hφ] ,k , Hφ −1 = AA T , where the latter identity uses that A is an orthonormal frame for Hφ.Hence we may rewrite Γ k = AA T [Hφ] ,k .Thus equation ( 14) is equivalent to: Because Λ has non-zero positive entries if its anti-commutator with a matrix is diagonal, then the matrix must be diagonal.The conclusion is that property C is equivalent to: By item (2) in Proposition 4.0.2 this is exactly the symmetry of the Christoffel matrices.
We can verify that strictly convex functions with orthogonal characteristics satisfy property C.
Furthermore, the following conditions are equivalent: (1) The image Hφ(Ω) is contained in the stratum Θ κ P .
(2) Property CK holds for all orthonormal frames in C over all points of Hφ(Ω).
In either case φ restricts to the leaves of the (parallel) foliation determined by the (rotation of ) the subspace d κ to a multiple of the standard quadratic form (up to an affine summand).
Proof.Let x and C be a point and orthonormal frame for Hφ(x) with respect which condition CK is defined.Because Ω is (path) connected for every point y we have a curve γ starting at x and ending at y.By hypotheses the horizontal lift of Hφ(γ) is a curve A(t) contained in C. Therefore it is in T C and horizontal.By (2) in Proposition 5.0.2A(t) is contained in the leaf CD of F : A(t) = CD(t).Because A(t) is a curve of orthogonal frames A(t) T Hφ(γ)A(t) = I.Therefore Equivalently, if C = BΛ then by (2) in Proposition 5.0.3 π| C sends the leaf CD diffeomorphically onto Ad B (D), which is where the image of Hφ must be confined.The (local) splitting condition for Hφ along the characteristics of C can be also argued as follows: that A(t) ⊂ CD implies that line fields at x spanned by each column of C are invariant by parallel transport.Thus the de Rham Splitting Theorem applies (and Hessian metrics restrict to Hessian metrics).
If Hφ ⊂ Θ κ P , then by item (1) in Proposition 5.0.4 the frame C ∈ C with respect to which CK is defined belongs to Θ κ C .By item (2) in the same Proposition all horizontal curves based at C must be contained in the leaf CΘ κ D of the foliation of Θ κ C induced by F upon clean intersection.The principal SO(n) κ -action takes these horizontal curves at C to horizontal curves in Θ κ C based at any matrix in the fiber.Conversely, let C = BΛ and assume that the horizontal lifts at CB ′ , B ′ ∈ SO(n) Λ , are contained in C. Then they are inside the corresponding leaf of F and therefore Because the exponential intertwines the adjoint action the latter intersection can be understood in s.If Λ ∈ d κ we have We used property CK with respect to an arbitrary point x ∈ Ω.If we select the point whose image lies in the stratum of smallest dimension, then we conclude that Hφ(Ω) cannot leave that stratum.
If Hφ ⊂ Θ κ P , then at any point x ∈ Ω all orthonormal frames in SO(n) κ are parallel.This means that the common eigendirections Ad B (d κ ) are parallel.Therefore the restriction of Hφ to the foliation given by the parallel translates of Ad B (d κ ) in Ω is flat.Hence on each such (affine) subspace it is a quadratic form with equal eigenvalues.Therefore in the local splitting of φ along orthogonal characteristics we shall have a multiple of the standard quadratic form along Ad B (d κ ).
The following result is more general than Theorem 1.0.2 in the Introduction.
Theorem 6.0.5.If A Hessian metric Hφ on Ω has property C and Hφ(Ω) is contained in a stratum of Θ P , then it has property CK.In particular φ has orthogonal characteristics.
Proof.Let Hφ(Ω) be contained in Θ κ P .This stratum is foliated by SO(n)Θ κ D and we want to show that for each curve γ(t) in Ω the derivative of Hφ(γ(t)) is tangent to this foliation.By property C for each t 0 there exists an orthogonal frame C ∈ C such that the horizontal curve A(t) at C has derivative at zero tangent to C: C is clean and A ′ (0) belongs to Θ κ C .Because the vector is horizontal by item (2) in Proposition 5.0.4 it is tangent to the foliation SO(n)Θ κ D .Thus by item (3) the tangent vector of Hφ(γ) at t 0 is tangent to the foliation SO(n)Θ κ D .Because Ω is connected this implies that Hφ(Ω) is contained in one of the leaves of SO(n)Θ κ D .The following result is a more precise statement that Theorem 1.0.3 in the Introduction: Theorem 6.0.6.Let Hφ be a real analytic Hessian metric on Ω ⊂ R n .Then φ has property C if and only if it has property CK.In such case Hφ is the restriction to Ω of a product Hessian metric on a (rotated) cube.
Proof.The stratification Θ P has a finite number of strata.Therefore there exists one stratum Θ κ P whose pullback by Hφ has non-empty interior Ω ′ ⊂ Ω.By Theorem 6.0.5 and Theorem 6.0.4 In particular Hφ(Ω) must be contained in the real analytic submanifold Ad B (D) 4 .By Proposition 5.0.4 the restriction π| C : BD → Ad B (D) is a diffeomorphism from a horizontal submanifold.Therefore all lifts of curves in Ω at C ∈ Θ κ C are contained in BD ⊂ C, which is property CK.
The image of Ω by the orthogonal projection onto a characteristic line is connected, and hence an interval.The restriction of φ to the foliation of Ω by affine lines parallel to the characteristic line is locally projectable.Because the interval has trivial topology local projections must agree on overlaps.

Bifurcation of orthogonal characteristics along quadratic forms
We shall construct a family of Hessian metrics with property I which do not have orthogonal characteristics, and, hence, by Theorem 6.0.4 do not have property CK.The family, despite being defined on domains which are not necessarily convex, will be also invariant under Legendre transform.
Firstly, we shall describe the domains Ω we are interested in.Let Ω 0 be an (open convex) polytope.By this we mean a domain defined as the points where a finite number of affine maps are strictly positive; the zero set of each such map is a supporting hyperplane.The closure of the polytope need not be compact.Let , respectively; we shall refer to R 1 l as the primary characteristic of Ω 1 l .We shall assume that R 1 l is oriented and we shall denote by p l the infimum of the interval I l (the interval may not be bounded from above).we shall refer to H l := p l × R n−1 l as the primary supporting hyperplane.We shall assume that • the polytopes Ω 0 , Ω 1 1 , . . ., Ω 1 m are disjoint and that Ω 1 i and Ω 1 j , i = j, have disjoint closure; • the primary supporting hyperplane H l for Ω 1 l is also a supporting hyperplane for Ω 0 and ∂Ω 1 l ∩ H l ⊂ ∂Ω 0 ∩ H l .We define Ω to be We refer to Ω as in (15) as a polytope with 1-handles.Secondly, we shall introduce appropriate strictly convex functions on the polytope with 1-handles.Let φ 0 be a multiple of the standard quadratic form on R n : ).Let y = (y 1 , . . ., y n ) be the coordinates which correspond to the image by B l of the canonical basis e 1 , . . ., e n and let q l (y) = q l (y 2 , . . ., y l ) = k(y Proposition 7.0.1.Let φ be the function defined on the polytope with 1-handles Ω as follows: where φ l (y) = φ l (y 1 ) a strictly convex smooth function on I l tangent at p l to ky 2 1 at infinite order.Then it has the following properties: (1) It is a smooth and strictly convex function on Ω.
(2) It has property I.
(3) The image Hφ(Ω) ⊂ P is contained in the union of the closed stratum and the two open strata of lowest dimension of Θ P .
(4) If there are two 1-handles on which φ l is not a quadratic form and the primary characteristic are neither equal not perpendicular, then Hφ does not have property CK.
Proof.That φ is smooth and strictly convex is a consequence of ( 16) and of the definition of φ l .The restriction of φ to Ω 0 has property I; the restriction to each Ω 1 l has orthogonal characteristics given by the columns of B l .Therefore φ has property I on the closure of the union, which is Ω.
By construction Hφ sends Ω 0 to the closed stratum of Θ P , and thus so Ω 0 ; if φ l is not a quadratic form, then Hφ sends an open subset of Ω 1 l to the strata positive matrices with two eigenvalues so that one is simple: By Theorem 6.0.4 if Hφ has property CK then φ has orthogonal characteristics for some C ∈ C (or B ∈ SO(n)).On a 1-handle Ω 1 l with φ l different from a quadratic form φ has orthogonal characteristics exactly for all B l SO(n) κ D. The 1-handles Ω 1 i and Ω 1 j have primary characteristics which are neither equal nor orthogonal if and only if Proposition 7.0.2.Let Ω be a polytope with 1-handles and let φ ∈ C ∞ (Ω) as in Proposition 7.0.1.Then the following holds: (1) The Legendre map dφ on Ω is a diffeomorphism, its image Ω * is a polytope with 1-handles, and Ω 1 l * and Ω 1 l have the same primary characteristic.(2) φ * is a function as in Proposition 7.0.1.
Proof.Because φ is strictly convex dφ : Ω → R n is a local diffeomorphism.Because φ| Ω 0 is a quadratic form dφ(Ω 0 ) is another polytope.The restriction φ| Ω 1 l =I l ×F l decomposes as a sum of strictly convex functions φ l +q l .The subset dφ(Ω 1 l ) is another 1-handle because I l is 1-dimensional, F l is a polytope and q l is a quadratic form; furthermore, because the Legendre transform commutes with orthogonal transformations dφ(Ω 1 l ) = I * l × F * l where the product decomposition is also with respect to R 1 l ×R n−1 l .The condition on the non-overlap of the closures of the 1-handles can be restated as follows: if two 1-handles have common primary supporting hyperplane, then their polytopes there have non-intersecting closure.This implies that if we prolongue each I l ⊂ R 1 l across p l to a larger interval Ĩl so that Ĩl ×F l ⊂ Ω 0 ∪p l ×F l ∪Ω 1 l , then Therefore dφ : Ω → dΩ is a bijection and thus a diffeomorphism onto another polytope with 1-handles.
Because the Legendre transform of a multiple of the standard quadratic form is a multiple of the standard quadratic form it follows that φ * belongs to the class of functions described in Proposition 7.0.1.On a toric variety a T-invariant Poisson structure has a simple infinitesimal description: its restriction to the open dense orbit -which upon fixing a point is identified with T -followed by the logarithm map, defines a constant Poisson structure in the Lie algebra of T. The infinitesimal counter part of a toric Poisson structure is a Hermitian inner product.If it is totally real it corresponds to an inner product on it, where t denotes the Lie algebra of T .In such case, we say that e 1 , . . ., e n ∈ it is an adapted Darboux basis if the inner product becomes standard; equivalently, e 1 , . . ., e n , ie 1 , . . ., ie n is a Darboux basis for the inverse constant symplectic structure.
On a toric variety a Hamiltonian Kahler form can be described by (Legendre dual) strictly convex functions: a Kahler potential in logarithmic coordinates and a symplectic potential in momentum map coordinates [6].
The Poisson commuting equation for a totally real Poisson structure and a Hamiltonian Kahler form corresponds -in appropriate coordinates -to property I: Theorem 8.0.3.Let (X, T) be a toric variety endowed with a totally real toric Poisson structure Π and a Kahler structure σ for which the action of T is Hamiltonian.Let P be the inverse Poisson structure of σ.Then the following statements are equivalent: (1) Π and P Poisson commute: [Π, P ] = 0.
(2) In an adapted Darboux basis the Kahler potential φ has property I.
(3) In an adapted Darboux basis the symplectic potential φ * has property I.
Proof.We regard the equation [Π, P ] = 0 as the defining equation for degree 2cocycles in the Poisson cohomology of Π: d Π P = 0.In logarithmic coordinates exp * Π has an inverse which is a (constant) symplectic structure Ξ on t ⊕ it.Therefore is an isomorphism of chain complexes (see e.g.[5, Proposition 6.12]).Hence Let e 1 , . . ., e n , ie 1 , . . ., ie n be an adapted Darboux basis for the totally real toric Poisson structure.In the fixed coordinates and associated frames of the complexified tangent and cotangent bundles the matrices of Π and Ξ are: denote the matrix of exp * P # , then Its exterior derivative is: Because the entries of g only depend on x we have: and g −1 is the Hessian of the Kahler potential φ.In particular g and its inverse are symmetric matrices.Renaming the set of indices in the first summand and using the symmetry of g we obtain: This is exactly property I for the Hessian metric g −1 = Hφ.The symplectic potential of σ is the Legendre transform of φ.By Proposition 4.0.4φ (in R n ) has property I if and only if φ * (in dφ(R n )) has property I. Theorem 8.0.4.Let (X, T) be a projective toric Poisson structure endowed with a toric Poisson structure Π which Poisson commutes with a real analytic Kahler structure σ for which the action of T is Hamiltonian.Then (X, T) is a Cartesian product of projective lines and both Π and σ factorize.
Proof.Because σ is real analytic the Kahler potential φ is real analytic; the Legendre transform preserves analytic (strictly convex) functions.Therefore by Theorem 8.0.3 the symplectic potential φ * has property I.By Theorem 6.0.6 φ * is defined in a Cartesian product of intervals I 1 × • • • × I n (we may dispense with the rotation by changing accordingly the adapted Darboux basis).One must have the equality dφ(R n ) = I 1 × • • • × I n because otherwise by repeating the Legendre transform we would get a domain for the original Kahler potential strictly containing R n .Thus the interior of the moment polytope ∆ is a Cartesian product of intervals.A product of intervals has a property invariant under affine transformations: it is limited by pairs of parallel hyperplanes.Because ∆ is a Delzant polytope there is an affine transformation that takes the integral lattice of t * to Z n , a vertex of ∆ to the origin and the facets containing this vertex to the coordinate hyperplanes.Because ∆ must be still described by parallel hyperplanes, is it actually a Cartesian product of intervals in this integral affine coordinates of t * .The fan of the Delzant polytope determines the toric variety (X, T).The fan of a cube in To show that the Kahler form σ also splits as a sum of Kahler forms on each projective line we use toric charts for (X, T).For that we observe that the linear part of the above affine transformation must be a permutation followed by a (signed) re-scaling of each Euclidean direction.Therefore if we dispense the affine transformation we deduce that the in the fixed compatible Darboux basis the subset ie 1 , . . ., ie n is -up to re-scaling of its members -an integral basis of t.Let us re-scale to a basis ǫ 1 , . . ., ǫ n , iǫ 1 , • • • , iǫ n so that the second block is an integral basis of it.In the corresponding coordinates the Kahler potential of σ is still a sum of strictly convex functions on each coordinate and therefore the Legendre diffeomorphism still sends it ∼ = R n to a cube.Let v be the vertex of its closure whose coordinates are smaller than those of the others.The basis of inner pointing integral vectors normal to the facets containing v is exactly iǫ 1 , • • • , iǫ n .Therefore for the standard toric chart associated to v (see e.g.[1, Chapter2, Section 5]) the identification 5 of T with (C * ) n comes from the Lie algebra identification which sends iǫ 1 , . . ., iǫ n to the canonical basis of R n ⊂ C n .In other words, upon having identified T with the open orbit of X, the product structure induced by X ∼ = CP 1 × • • • × CP 1 on T is exactly the factorisation of the torus coming from the (complex) basis e 1 , • • • , e n of its Lie algebra.The factorisation of X decomposes because it has that property in the open dense subset T. The compatibility of Π with the product structure is also immediate.Therefore 5 Though we do not need it here, the toric chart could be chosen compatible with the monoid structure, so that (1, . . ., 1) corresponds to the fixed point in the open orbit.
Firstly, we show how upon adding the orbit closures to exp(Ω 1 l * ⊕ iR n ) (and not in exp(Ω * ⊕ iR n )) we get an open subset X l ⊂ X which is T -invariant and to which the product structure in (17) extends.For that we use the toric atlas (as a monoid) determined by the polytope ∆ ([1, Chapter2, Section 5]): To each vertex v ∈ ∆ there correspond a toric chart which identifies the union of orbits of X which correspond to the star of v with the standard affine toric variety: (C n , (C * ) n ).The standard integral basis of the Lie algebra of (S 1 ) n comes from the integral linear forms ν l1 , . . ., ν ln associated to the affine forms α lj ; in particular this describes how for each toric chart t ⊕ it -for which we already have picked a basis -is identified with the Lie algebra of the standard complex torus R n ⊕iR n .The point in the open orbit of X which determines the monoid structure goes to the unit (1, . . ., 1) ∈ C n .Let us fix a toric chart of a vertex v which belongs to the supporting hyperplane of Ω 1 l and ∆.Under the identification of T with the standard torus (C * ) n we can assume that the (trivialized) subgroup C * l ∼ = C * maps to the first factor of the standard torus so that on trivializations the isomorphism is given by the inversion.Thus D o l maps to a semigroup D i l ⊂ C * contained in the unit disk.Let W l ⊂ C n be the image in the toric chart of the second factor exp(F l ⊕ iR n−1 l ).Then the image of exp(Ω 1 l * ⊕ iR n ) is (zw 1 , . . ., w n ), z ∈ D o l , w = (w 1 , . . ., w n ) ∈ W l .Therefore the image of X l is also completely contained in the toric chart and equals (zw 1 , . . ., w n ), z ∈ D o l ∪ {0}, (w 1 , . . ., w n ) ∈ W l .Because W l is a codimension 2 submanifold which intersects each complex line parallel to the z 1 -axis transversely in at most one point, we deduce that X l is an open subset which extends the product structure.By construction it is also T -invariant.
To show that σ extends to a Kahler form on X l we shall work with its inverse Poisson structure P .The decomposition φ * | Ω 1 l * (y) = φ * l (y 1 )+q * l (y 2 , . . ., y 1 ) implies that on the exponentiation of the 1-handle P decomposes as P l + P ′ l , where each summand is a field of bivectors tangent to one of the foliations in the product decomposition (17).The second field of bivectors is easier to describe: In the Lie algebra the foliation is given by translates of F l ⊂ R n−1 l .On each such leaf the Kahler potential for the corresponding Kahler form is the quadratic form q l .Therefore P ′ l corresponds to a constant bivector on Ω 1 * × iR n .The exponentiation of a constant bivector to the (abelian) Lie group has an alternative description: it is the field of bivectors obtained by replacing each vector in the decomposition in ∧ 2 (t ⊕ it) by its corresponding infinitesimal vector field for the action by (left) multiplication.Because the action of T on itself extends to an action on X it follows that P ′ l is the restriction of a (Poisson) structure on X.To describe P l on each semigroup orbit we may assume that φ l (y 1 ) equals − For boundary components in the inner boundary we change coordinates by a rotation so that all supporting hyperplane involved are coordinate hyperplanes.Then we impose the same boundary conditions as above on the corresponding summands of the multiple of the standard quadratic form in these coordinates.We may have supporting hyperplanes with non-empty intersection, say k of them.This means that we shall have work with the corresponding coordinates and hence with a splitting into a vector subspace of dimension k and its orthogonal complement.We shall work on a toric chart associated to a vertex in the intersection of the supporting hyperplanes.There, the foliation corresponding to the vector subspace will have leaves given by the action of (C * ) k on the first k coordinates on an appropriate slice.Hence by adding closure of (semigroup) orbits we shall obtain an open subset.The computation of the Kahler potential for the inverse symplectic form of P l on such leaves is analogous.
To computation of the image of the momentum map is straightforward.
The proof of Proposition 1.0.9 in the Introduction is a minor variation of the following: Example 8.0.6.(Attaching a toric 1-handle to the standard commuting pair) Let ∆ be the standard n-simplex in R n .Let Ω 0 be the truncation of the (open) cube of side (0, 1 n ) by the hyperplane x 1 + • • • + x n = 2 3 .Let Ω 1 be the 1-handle with the primary characteristic spanned by (1, • • • , 1), and so that its primary supporting hyperplane is 3 , the parallel one is x 1 + • • • + x n = 1 and the n − 1-dimensional polytope is the (translation of) the intersection of the cube and the primary supporting hyperplane.
We let Ω be the polytope with 1-handles determined by Ω 0 and Ω 1 above.It is in the hypotheses of Theorem 8.0.5.By going through the its proof we check that: • In the toric chart associated to the origin σ (near the origin) will be the standard (constant) Kahler form i 2 j dz j ∧ dz j ; the open cube is a (punctured) polydisk which is appropriately truncated.
• Near the truncation hypersurface W the Kahler form is i 2 j 1 zj zj dz j ∧ dz j .• Attaching the 1-handle amounts to the following: the truncating hypersurface W is stable under the diagonal action of S 1 .Then each such orbit is being 'capped' by a (holomorphic) disk which is Kahler for σ; the disk is nothing but (a part of) the projective line determined by the orbit, its center being in the hyperplane at infinity CP n = C n ∪ CP n−1 ; this is done for the whole F 1 × (S 1 ) n−1 -family.

Proposition 1 . 0 . 6 .
The restriction of the universal Levi-Civita connection to C defines a (regular) involutive distribution.Its leaves are the left translates of the strictly positive matrices D which fit in the Cartan factorization C = SO(n)D.
(a) Every strictly convex function of one variable has property I.(b) If φ 1 (x 1 ), . . ., φ n (x n ) are strictly convex, then φ 1 (x 1 )+• • •+φ n (x n ) has property I in the product of the corresponding intervals.(c) If φ(x) has property I in Ω and B ∈ O(n) is an orthogonal transformation, then φ(Bx) has property I in B −1 (Ω).This is because Hφ(Bx) = B T Hφ(x)B

Lemma 2 .0. 2 .
If φ ∈ C ∞ (Ω) has orthogonal characteristics, then φ is an strictly convex function with property I.More precisely, φ is the composition of an orthogonal transformation with a function with trivial mixed partial derivatives.Proof.Because C has orthogonal columns we can factor C = BΛ, Λ ∈ D, B ∈ SO(n).From L 2

Definition 4 .
0.1.A Hessian metric Hφ on Ω has symmetric Christoffel symbols if the Christoffel symbols (of the second type) are symmetric on the three indices.Equivalently, if its Christoffel matrices are symmetric.

Proposition 4 . 0 . 2 .
The following statements for a Hessian metric Hφ on Ω are equivalent: (1) It has symmetric Christoffel symbols.(2) The two matrices Hφ and [Hφ] ,k are in the same Cartan subalgebra of the symmetric matrices, 1 ≤ k ≤ n.(The Cartan subalgebra may vary with k).(3) The two matrices Hφ and Γ k are in the same Cartan subalgebra of the symmetric matrices, 1 ≤ k ≤ n.(The Cartan subalgebra may vary with k).
(a) A function φ ∈ C ∞ (Ω) has property I if and only if Hφ has symmetric Christoffel symbols.The Christoffel symbols are the components of the Levi-Civita connection.Therefore one may expect a reformulation of property I related to the tangent or orthogonal frame bundle with the Levi-Civita connection.(b) If a function φ ∈ C ∞ (Ω) has orthogonal characteristics, then the Hessian metric

Proposition 5 .0. 2 .
The subset of matrices with orthogonal columns C ⊂ Gl(n) + has the following properties:(1) It is a closed embedded submanifold of Gl(n) + on which the Cartan decomposition Gl(n) + = SO(n)P induces a product structure C = SO(n)D.(2)The intersection of the horizontal distribution of ∇ with the tangent bundle T C is an involutive distribution on C. Its foliation F is the one associated to the Cartan decomposition, with leaves the left SO(n)-translates of D.
At C the respective tangent spaces are C • d and C • d κ .Therefore the intersection is clean and this proves (2).By item (1) above and by item (2) in Proposition 5.0.3 the restriction of π| Θ κ C to the leaf C • Θ κ D ⊂ Θ κ C is a diffeomorphism over its image.Its image is Ad B (Θ κ D ) (π| D : D → D is the inverse of the square map); it is in fact the common image of all leaves through points of the fiber CSO(n) κ .We can now sharpen Proposition 4.0.2.Proposition 5.0.5.Let Hφ be a Hessian metric on Ω such that Hφ(Ω) is contained in the stratum Θ κ P .Then its Christoffel symbols are symmetric if and only if Hφ and [Hφ] k can be conjugated by an orthogonal matrix to a matrix in d κ , 1 ≤ k ≤ n.Proof.Because the image of Hφ is contained in Θ κ P , its partial derivatives must be in the tangent space to the stratum: [Hφ] k ∈ T Θ κ P .By item (2) in Proposition 4.0.2there exists a special orthogonal matrix B which conjugates Hφ and [Hφ] k to a diagonal one: B T [Hφ] k B ∈ d.Therefore B T [Hφ] k B ∈ d ∩ T Θ κ P = d κ .

Lemma 6 . 0 . 2 .
If L 2 C φ ∈ D, C ∈ C, then Hφ satisfies property C. Proof.By definition C T HφC = Λ, Λ ∈ D. Therefore C T [Hφ] ,k C ∈ d.The matrix CΛ −1/2 also belongs to C and it is an orthonormal frame for Hφ.Therefore (CΛ −1/2 ) T [Hφ] ,k CΛ −1/2 ∈ d, and thus property C holds.Next we analyze up to which extent Hessian metrics with property C are defined by functions with orthogonal characteristics.Definition 6.0.3.A Hessian metric Hφ on Ω ⊂ R n has property CK if there exists a point x ∈ Ω and an orthonormal frame C ∈ C for Hφ(x), such that for every curve in Ω based at x its horizontal lift at C is contained in C. Theorem 6.0.4.If a Hessian metric Hφ on Ω has property CK, then it solves

8.
An application to Poisson geometry We shall show that property I for strictly convex functions condition is equivalent to the Poisson commuting equation for Poisson structures related to Kahler forms on toric varieties.We shall use (a) our classification of real analytic inversible Hessian metrics to deduce a factorization result; (b) the family of strictly convex functions with property I introduced in Section 7 to produce pencils of Poisson structures on regions of projective varieties which interpolate from a Kahler structure to a Poisson structure with a finite number of Kahler leaves.Definition 8.0.1.[2, Section 4] A Poisson structure Π on a toric variety (X, T) is (1) toric if the bivector field Π is T-invariant, of type (1, 1) and positive, and if the symplectic leaves of Π equal the finitely many orbits of the torus action; (2) totally real if the orbits of the (maximal) compact torus T are coisotropic submanifolds.Remark 8.0.2.Totally real toric Poisson structures are good candidates to be limits of Hamiltonian Kahler forms: for such a form its inverse Poisson bivector isT -invariant, of type (1, 1) and positive; there is a unique symplectic leaf of which the T -orbits are Lagrangian submanifolds.Thus it is natural to look for converging sequences of such bivectors so that in the limit the unique symplectic leaf breaks into the finitely many orbits and the T -symmetry is enlarged to T-symmetry.One possible source would be a totally real toric Poisson structure which Poisson commutes with (the inverse of) a Hamiltonian Kahler form.In such case the convex combination of the bivectors would be a smooth family of (inverses of) Hamiltonian Kahler forms converging to the totally real toric Poisson structure.

1 2 α 2 1z 1
l (log(α l − 1) everywhere in I l .The Legendre transform of − 1 2 (−r)(log(−r) − 1) for r < 0 is 1 2 exp −2r .Using that the Legendre transform commutes with orthogonal transformations and its behavior under translation and scaling we obtain φ * l (y 1 ) = 1 2 exp −2y1/|ν l | −y 1 d l , where d l is the distance of the boundary point of I l different from p l .Since we are interested in Kahler forms/bivectors we may dispense with the linear summand.Under the semigroup identification log : D o l → I * l ⊕ iR the potential pull backs to 1 2z z .Under the identification D o l → D i l it maps to 1 2 z z.Under the action on the standard toric chart it maps to 1 2a z1 .Hence the bivector P 1 there equals ∂ z1 near z 1 = 0, which extends to X l .Both P l and P ′ l are non-degenerate in the added points, and thus σ extends to a Kahler form.
both Hφ and [Hφ] ,k are in Ad C (d).By item (2) in Proposition 4.0.2 the same occurs for Γ k (the action by conjugation on d of second factor of C = SO(n)D is trivial).By Proposition 4.0.2 this implies symmetry of Christoffel matrices.
Proposition 4.0.4.The Legendre transform preserves the class of Hessian metrics with property I on convex domains.Proof.Let L j denote the Lie derivative with respect to ∂ ∂xj .We can rewrite property I for Hφ as

•
The inverse of Kahler form σ is a Poisson bivector field P which Poisson commutes with the totally real toric Poisson bivector field Π which in the previous toric chart is2