Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics

We provide an easy approach to the geodesic distance on the general linear group GL(n) for left-invariant Riemannian metrics which are also right-O(n)-invariant. The parametrization of geodesic curves and the global existence of length minimizing geodesics are deduced using simple methods based on the calculus of variations and classical analysis only. The geodesic distance is discussed for some special cases and applications towards the theory of nonlinear elasticity are indicated.


Introduction and preliminaries
The interpretation of the general linear group GL(n) as a Riemannian manifold instead of a simple subset of the linear matrix space R n×n has recently been motivated by results in the theory of nonlinear elasticity [22,21], showing a connection between the logarithmic strain tensor log √ F T F of a deformation gradient F ∈ GL + (n) and the geodesic distance of F to the special orthogonal group SO(n). Since the requirements of objectivity and isotropy strongly suggest a distance measure on GL + (n) which is right-invariant under rotations and left-invariant with respect to action of GL + (n), we restrict our considerations to Riemannian metrics on GL(n) which are left-GL(n)-invariant as well as right O(n)-invariant.
Although the theory of Lie groups is obviously applicable to GL(n) with such a metric, this general approach utilizes many intricate results from the abstract theory of differential geometry and is therefore not easily accessible to readers not sufficiently familiar with these subjects. Furthermore, while the explicit parametrization of geodesic curves has been given for the canonical left-invariant metric on GL + (n) [1] as well as for left-invariant, right-SO(n)-invariant metrics on SL(n) [18], analogous results are not found in the literature for the more general case on GL(n).
The aim of this paper is therefore to provide a more accessible approach to this type of Riemannian metrics on GL(n) and the induced geodesic distances as well as to deduce the parametrization of geodesic curves using only basic methods from the calculus of variations and classical analysis. In order to keep this article as self-contained as possible, we will begin by stating (and proving) some very basic facts on Riemannian metrics for the special case of GL(n).

Basic Definitions
Let R n×n denote the set of all n × n real matrices and let 1 denote the identity matrix in R n×n . We define the groups GL(n) = {X ∈ R n×n | det(X) = 0} (general linear group) , Furthermore, we define the set of symmetric matrices Sym(n) = {X ∈ R n×n | X T = X}, the set of positive definite symmetric matrices Sym + (n) = {X ∈ R n×n | X T = X and v T Xv > 0 ∀ v ∈ R n \ {0}} and the set of skew symmetric matrices so(n) = {X ∈ R n×n | X T = −X}. While dist Euclid induces a distance function on GL(n) as well, it does not appear as a "natural" inner property of the general linear group: since GL(n) is not a linear space, the term A − B depends on the underlying algebraic structure of the vector space R n×n . Furthermore, because GL(n) is not a closed subset of R n×n , it is not complete with respect to the Euclidean distance.

Distance functions on R n×n
A more proper distance measure should take into account the algebraic properties of GL(n) as a group. To find such a function we interpret GL(n) as a Riemannian manifold: as an open subset of R n×n , the tangent space T A GL(n) at an arbitrary point A ∈ GL(n) is given by 1 T A GL(n) = A · T 1 GL(n) ∼ = A · R n×n = R n×n =: gl(n) . (1.5) To obtain a distance function respecting this structure on GL(n), we will consider a measurement along connecting curves.

The geodesic distance
A Riemannian metric on GL(n) is a smooth (i.e. infinitely differentiable) function g : GL(n) × gl(n) × gl(n) → R, (A, M, N ) → g A (M, N ) (1.6) such that for every fixed A ∈ GL(n) the function g A (·, ·) : gl(n) × gl(n) → R is a positive definite symmetric bilinear form, i.e. where we employ the notationẊ(t) = d dt X(t). Note that the length of X is defined similarly to the length of curves in Euclidean spaces. Thus many wellknown properties, like invariance under reparameterization, still hold in the Riemannian case. Some such properties will be discussed further in the following section.
The geodesic distance dist geod (A, B) between A, B ∈ GL(n) can now be defined as the infimum over the length of curves connecting A and B. For this we need an exact definition of the admissible sets of curves. the set of regular k-times differentiable curves in GL(n) over the interval [a, b]. Note that, by the usual definition of differentiability on closed intervals as the restriction of differentiable functions on R, the i-th derivative X (i) (a) and X (i) (b) at the boundaries is well-defined via the one-sided limits lim t a X (i) (t) and lim t b X (i) (t). We now define the set of piecewise k-times differentiable curves in GL(n) over the interval [a, b] by ; GL(n)) | ∃ a = a 0 < a 1 < · · · < a m+1 = b, ∀ j = 0, . . . , m : Note that, by this definition, partially differentiable curves are continuous everywhere. Finally, for A, B ∈ GL(n), the admissible set of curves connecting A and B is and the general admissible set of curves is While the notion of piecewise differentiability is often found in the literature, a specific definition is sometimes omitted. The definition used here guarantees the existence of one-sided limits lim t t0 X (i) (t) everywhere and thus, in particular, that the length L(X) < ∞ is well-defined.
We can now properly define the geodesic distance function: is called the geodesic distance between two matrices A and B.
Remark 1.4. It is easy to verify that dist geod is indeed a distance function; in order to see that it satisfies the triangle inequality (1.1), choose a curve X ∈ A B A connecting A and B with L(X) ≤ dist geod (A, B) + ε as well as a curve Y ∈ A C B connecting B and C with L(Y ) ≤ dist geod (B, C) + ε. We assume (without loss of generality, as we will see in Lemma 1.5) that both X and Y are defined on the interval [0, 1]. Let Z denote the curve obtained by "attaching" Y to X, i.e. . (1.14) Then Z is piecewise differentiable, Z(0) = X(0) = A and Z(2) = Y (1) = C, and thus Z ∈ A C A . We find 2 and thus dist geod (A, C) ≤ L(Z) ≤ dist geod (A, B) + dist geod (B, C) + 2ε for all ε > 0, which shows the triangle inequality.
The geodesic distance can be considered a generalization of the Euclidean distance: if we measure the length of a curve γ : [a, b] → R n×n by L(γ) = b a γ,γ dt, then the shortest curve connecting A, B ∈ R n×n is a straight line with length A − B . Thus the Euclidean distance can be interpreted as the infimum over the length of connecting curves as well.
Furthermore, GL(n) is not connected, but can be decomposed into two connected components GL + (n) = {A ∈ GL(n) | det A > 0} and GL − (n) = {A ∈ GL(n) | det A < 0}. Thus dist geod (A, B) < ∞ if and only if A and B are in the same connected component, i.e. iff det(AB) > 0. As we will see later on, for left-invariant Riemannian metrics we can focus on the case A, B ∈ GL + (n) without loss of generality.

Length and energy of curves
In order to further investigate the geodesic distance, some basic properties of curves in GL(n), the length and the energy functional are required. Some of these properties can be found in any textbook on differential geometry; however, in order to keep this article self contained and accessible to readers unfamiliar with the methods of general differential geometry, we explicitly state and prove them here. Many properties of curves in GL(n) also correspond directly to the case of curves in the Euclidean space (see e.g. [13,Chapter 12]).
Consider the integrand g X(t) (Ẋ(t),Ẋ(t)) in the definition of the length of a curve X. In analogy to the Euclidean space, we call this term the speed of X. Since, by definition of a Riemannian metric, g A (·, ·) defines an inner product on the tangent space T A GL(n) = R n×n = gl(n), it induces a norm on gl(n). We will therefore simplify notation by writing M A = g A (M, M ) for A ∈ GL(n), M ∈ gl(n). Then a differentiable curve X ∈ A([a, b]) has constant speed with regard to the Riemannian metric g if the mapping Proof. See Lemma A.3 in the appendix.
the (unique) reparametrization from Lemma 1.5. An important property of the length functional is its invariance under reparametrizations: ) be a piecewise continuously differentiable function with ϕ(c) = a, ϕ(d) = b and ϕ (t) > 0. Then Proof. See Lemma A.4 in the appendix.
To explicitly compute the geodesic distance, we will primarily search for length minimizers, i.e. curves X ∈ A B A which satisfy  If we are interested only in the length of a curve X, we will therefore often assume without loss of generality that X is defined on [0, 1] and that X has constant speed.
However, the energy functional is not invariant under reparameterization: For a given curve X : [0, 1] → GL(n) and λ > 0, the energy of the curve Y : For λ → ∞, we see that by admitting arbitrary parametrizations, the infimal energy of curves connecting A, B is zero whenever A and B can be connected: (1.20) We will therefore call X : an energy minimizer if and only if it minimizes the energy over all curves over the same parameter interval connecting A and B: The next two lemmas show important relations between length minimizers and energy minimizers. 22) and equality holds if and only if X has constant speed. Proof.
and X has constant speed.
Proof. Recall that X c denotes the (unique) parameterization of X with constant speed on [a, b]. The invariance of the length under reparameterization implies L(X c ) = L(X), and inequality (1.22) from the previous lemma yields where, again, equality holds if and only if X has constant speed, i.e. X = X c . Therefore, every minimizer of E must have constant speed, and it remains to show that if X has constant speed, then the equivalence holds. If we assume that X = X c , then as well as which concludes the proof.

Left-invariant, right-O(n)-invariant Riemannian metrics
In the following, we will only consider Riemannian metrics that are left-invariant as well as right-invariant under O(n). The left-invariance of a Riemannian metric g can be applied directly to the geodesic distance: let A, B, C ∈ GL(n). Then for every given curve X ∈ A B A connecting A and B we can define a curve Y = CX ∈ A (CB) (CA) connecting CA and CB by Y (t) = CX(t). We assume without loss of generality that X is defined on the interval [0, 1] and find Analogously, for every curve Y connecting CA and CB, the curve X = C −1 Y connects A and B with L(X) = L(Y ). Thus for every curve connecting A and B, we can find a curve of equal length connecting CA and CB and vice versa. Therefore (1.31) We will therefore often focus on the case A = 1. Since A −1 B ∈ GL + (n) for A, B ∈ GL − (n), the geodesic distance on GL − (n) is completely determined by the geodesic distance on GL + (n) for left-GL(n)-invariant metrics.
for all A ∈ GL(n), M, N ∈ gl(n) and Q ∈ O(n).
Such invariant metrics appear in the theory of elasticity, where right-O(n)-invariance follows from material isotropy, while objectivity implies the left-invariance. We will show that a Riemannian metric satisfying both invariances is uniquely determined up to three parameters µ, µ c , κ > 0 and given by an isotropic inner product ·, · µ,µc,κ of the form We will state some basic properties of ·, · µ,µc,κ . The proof can be found in the appendix (Lemma A.6). where Sym(n) and so(n) denote the sets of symmetric and skew symmetric matrices in R n×n respectively.
In particular, (1.36) implies that the canonical inner product can be interpreted as a special case of ·, · µ,µc,κ with µ = µ c = κ = 1. Furthermore we will denote by the norm induced by ·, · µ,µc,κ . Here and throughout, M = n i,j=1 M 2 ij denotes the canonical matrix norm.
We can now show the connection between the isotropic inner product and left-invariant, right-O(n)invariant Riemannian metrics.
where ·, · * is an inner product on the tangent space gl(n) = T 1 GL(n) at the identity 1.

ii) A left-invariant metric g is additionally right-O(n)-invariant if and only if g is of the form
with µ, µ c , κ > 0.
Proof. i) Let ·, · * be an inner product on gl(n) and g be defined by hence g is left-invariant. Now let g be an arbitrary left-invariant Riemannian metric on GL(n). Then defines an inner product on gl(n), and we find We apply the isotropy property (1.34) to find Finally, let g be an arbitrary left-invariant, right-O(n)-invariant metric on GL(n). Again we define the inner product ·, · * by M, N * := g 1 (M, N ). Then for every Q ∈ O(n) According to a well-known representation formula for isotropic linear mappings on R n×n [6], this invariance directly implies that ·, · * has the desired form (1.33).
In the following sections g will denote a left-invariant, right-O(n)-invariant Riemannian metric as given in (1.40), unless otherwise indicated.

Energy minimizers and the calculus of variations
In the theory of Lie Groups and, more generally, Riemannian geometry, it can be shown that every length minimizing curve on a sufficiently smooth Riemannian manifold is a geodesic [4, Corollary 3.9]. However, by focussing solely on the considered special case, we can circumvent the methods of abstract Riemannian geometry (such as the Levi-Civita-connection) and find a differential equation characterizing the minimizing curves by a straightforward application of the classical calculus of variations. A similar approach can be found in [15], where a geodesic equation similar to (2.8) is computed for the right-invariant Riemannian metric induced by the canonical inner product on gl(n). In preparation we need the following two lemmas. For any continuously differentiable function f ∈ C 1 (GL(n); GL(n)), we denote by Df [A] the total derivative of f at A ∈ GL(n), and Df [A].H ∈ gl(n) denotes its directional derivative at A in direction H. (2.1)

Proof.
A short proof can be found in [20].

The geodesic equation for the canonical inner product
First, assume that ·, · µ,µc,κ is the canonical inner product on ) be a two-times differentiable length minimizing curve from [0, 1] to GL + (n), where the length is measured by the left-invariant Riemannian metric g with and assume without loss of generality (Lemma 1.5) that X has constant speed. Then it follows from Lemma 1.8 that X also minimizes the energy functional E. We will characterize the minimizer X by the Euler-Lagrange equation corresponding to the energy, which we will call the geodesic equation, in consistence with a more general differential geometric definition of geodesics [12, Definition 1.4.2]. Let The minimizing property of X implies the stationarity condition

Thus, with integration by parts, (2.4) computes to
Since this equation holds for all V ∈ C 1 0 ([0, 1]; gl(n)), we can apply the fundamental lemma of the calculus of variations to obtain the differential equation We compute the left hand term (using the product rule for matrix valued functions [20] as well as, again, Lemma 2.1): To simplify notation we define U := X −1Ẋ . Theṅ and combining (2.5) with (2.6) we obtain Vandereycken et al. [25] give an analogous equation for the case of the canonical right-invariant metric.

The geodesic equation for the general metric
Let us now consider the general case of arbitrary µ, µ c , κ > 0. In order to find the geodesic equation for this case, we need the following lemma.
Proof. Since tr(AB) = tr(BA), we immediately see that which completes the proof.
Let X ∈ A B A be a piecewise two-times differentiable energy minimizer with regard to the Riemannian metric Note that, in contrast to Section 2.1, we do not require X to be differentiable everywhere, but . . , m}. Furthermore, following an approach by Lee et al. [15], we consider a different variation: holds for the energy minimizer X. Again, define U := X −1Ẋ , as well as We therefore obtain Coming back to equation (2.14), we find that where a = a 0 < a 1 < · · · < a m+1 = b are chosen such that X [aj ,aj+1] ∈ C 2 ([a j , a j+1 ]; GL(n)) for all We can now apply the fundamental lemma of the calculus of variations to finḋ To show that X is two-times differentiable on [0, 1] and that equation (2.22) is satisfied everywhere, we first show that U = X −1Ẋ is continuous at each a k (and therefore on the whole interval [0, 1]). We choose V ∈ C 1 0 ((a k−1 , a k+1 ); gl(n)) for k ∈ {1, . . . , m} and compute According to (2.22) the equalityU holds on (a k−1 , a k ) as well as (a k , a k+1 ). Thus the integrals in (2.23) are zero and we find Therefore U is continuous on [0, 1]. But then (2.22) implies thatU is continuous as well. ThusẊ = XU is continuous on the whole interval, and therefore X is continuously differentiable. But thenẊ = XU is continuously differentiable as well, and thus X ∈ C 2 ([0, 1]; GL + (n)), and U = X −1Ẋ satisfies the geodesic equationU everywhere on [0, 1]. The results of Section 2.2 can be summarized as follows: Proposition 2.4. Let A, B ∈ GL + (n) and let X ∈ A B A be a piecewise two-times differentiable energy minimizer with regard to the left-invariant Riemannian metric induced by the inner product ·, · µ,µc,κ . Then X ∈ C 2 ([a, b]; GL + (n)), and U = X −1Ẋ is a solution to (2.26).
Remark 2.5. In particular, every length minimizing curve can be reparameterized to an energy minimizer, according to Lemmas 1.5 and 1.8. Thus every length minimizer has a reparameterization that satisfies the geodesic equation.
Remark 2.6. Equation (2.26) could also be deduced from a more general formula given by Bloch et al. [3], which is derived via the Euler-Poincaré equations corresponding to the length minimization problem [16, p. 430ff.].

Properties of solutions U
Some properties of the solutions U ∈ C 1 ([0, 1]; gl(n)) to (2.26) can be inferred directly from the equation.
Lemma 2.7. Let U be a solution to (2.26). Then:

30)
where Cof U denotes the cofactor of U .  iv) We first assume that U is invertible. Then and since det(U ) is constant as well, we find The general case follows directly from the density of GL(n) in gl(n) = R n×n .

Existence and uniqueness of solutions
Since the mapping U → 1+ µc µ 2 (U T U − U U T ) is locally Lipschitz continuous, the Picard-Lindelöf theorem implies that equation (2.26) has a unique local solution for given U (0) and that if a global solution exists, it is uniquely defined. Similarly, for given U ∈ C 1 (I, gl(n)) on an arbitrary interval I, the initial value problem Ẋ = XU , has a unique local solution as well. Using estimates (2.27) to (2.29), we can even show that this solution is globally defined: for the geodesic equation has a unique global solution X : R → GL + (n).
Proof. Since U (0) = X(0) −1Ẋ (0) is determined by X(0) andẊ(0), (2.36) has a unique local solution, as demonstrated above. Then, according to (2.27), U µ,µc,κ is constant, hence U is bounded in R n×n and therefore defined on all of R. The same estimate shows that the mapping X → U X is globally Lipschitz continuous. Therefore (after rewriting (2.36) 1 asẊ = U X) there exists a unique global solution X : R → R n×n to (2.36). It remains to show that X(t) ∈ GL + (n) for all t ∈ R. First we observe that, for det X > 0, Thus on every interval I ⊆ R with 0 ∈ I and X(I) ⊆ GL + (n), the function t → det X(t) solves the initial value problem det X(0) = det X 0 . (2.38) Assume now that X(R) GL + (n). Since I + := {t ∈ R | det X(t) > 0} is open and 0 ∈ I + by definition of X 0 , there is a minimal t 0 > 0 with −t 0 / ∈ I + or t 0 / ∈ I + . But then det X is the (unique) solution to (2.38), i.e. det X(t) = (det X 0 ) e t·tr(U (0)) (2.39) on (−t 0 , t 0 ), and thus in contradiction to the definition of t 0 .
3 The parameterization of geodesics

Solving the geodesic equation
We will now give an explicit solution to the geodesic initial value problem (2.36). The result is inspired by the work of Mielke [18], who obtained formula (3.3) as a parametrization of the geodesics on SL(n). For the canonical inner product, i.e. for the special case µ = µ c = κ = 1, the result can also be found in [1]. Note that while geodesics on a Lie group equipped with a bi-invariant Riemannian metric are translated one-parameter groups which can easily be characterized through the matrix exponential [24] (which, for example, allows for an easy explicit computation of the geodesic distance on the special orthogonal group SO(n) with respect to the canonical metric [19]), the metric discussed here is not bi-GL(n)-invariant and thus the solution to the geodesic equation takes on a more complicated form. Let A ∈ GL + (n), M ∈ gl(n) and ω = µc µ . We define 2) where exp : gl(n) → GL + (n) denotes the matrix exponential (cf. Appendix A). We use equation (A.12) to computeẊ where we used (A.8) stated in the appendix to infer Q ∈ SO(n). Then and thus X solves the geodesic equation. To solve the initial value problem (2.36) we use the equality exp(0) = 1 as well as equation (3.4) to obtain hence we can solve (2.36) by choosing A = X 0 , M = X −1 0 M 0 . We conclude: Theorem 3.1. Let X 0 ∈ GL + (n), M 0 ∈ gl(n). Then the curve X : R → GL + (n) with is the unique global solution to the geodesic initial value problem Because of the left-invariance of g, we can mostly focus on the case A = 1, Φ A = Φ 1 = Φ and

Properties of geodesic curves
Since the geodesic curves solve the geodesic initial value problem (2.36), the properties given in Lemma 2.7 can be directly applied to U = X −1Ẋ . Furthermore, we can compute the length of X on the interval [0, t 0 ] for given M ∈ gl(n) (recall from (3.5) that X −1Ẋ = U = Q T M Q with Q(t) ∈ SO(n)): where the second to last equality follows from the isotropy property (1.34) of the inner product ·, · µ,µc,κ .

The existence of length minimizers
As we have seen in Section 2 (Remark 2.5), every sufficiently smooth length minimizing curve can be reparameterized to solve the geodesic equation. Thus Theorem 3.1 shows that every length minimizer, after possible reparameterization, is of the form (3.3). At this point, it is not clear that, for given A, B ∈ GL + (n), such a minimizer connecting A and B actually exists. In the broader setting of differential geometry it can be shown that the local existence of minimizers is guaranteed on any (differentiable) Riemannian manifold [12,Corollary 1.4.2]. Furthermore, the global existence of the geodesic curves demonstrated in Proposition 2.8 implies that GL(n) is geodesically complete with respect to g, and thus the Hopf-Rinow theorem also guarantees the global existence of length minimizers. Nonetheless, in our effort to keep this paper self contained, we will give a full proof for the existence of minimizers using only results from basic real analysis. Towards this aim, we will first show that a specific variant of Gauss's lemma [8, p. 2.93] holds for g on GL(n) and continue by following the proof of the Hopf-Rinow theorem as given in [12, Theorem 1.7.1] (a similar proof can be found in [4, Theorem 2.8]) as applied to our special case. Readers not interested in these rather basic proofs should skip the main part of Section 4 and continue with Theorem 4.7.

Preparations
In the context of Riemannian geometry the function Φ A : gl(n) → GL + (n) with can be considered the geodesic exponential at A [12, Definition 1.4.3]. Note again that from this point of view the following lemma is a direct application of Gauss's lemma. We will prove it via direct computation.  This lemma can now be used to establish a lower bound for the length of curves "close to A". To do so, we choose ε > 0 such that Φ A is a diffeomorphism from B ε (0) ⊆ gl(n) to an open neighbourhood of A in GL + (n); note that, according to (3.8), DΦ A [0] is surjective and therefore nonsingular. Then any "short enough" curve Y with Y (0) = A can be represented as Y = Φ A • γ with a curve γ in B ε (0) ⊆ gl(n). To compute the length of Y , we need the following lemma, which is a modified version of Proposition 5.3.2 in [7]: Proof. Let such a curve γ be given. Then the length of Φ A • γ is Note that since γ andγ are generally not equal, the previous lemma can not be applied directly. Therefore we decomposeγ into the sum of ξ 1 and ξ 2 , where ξ 1 is a multiple of γ and ξ 2 is orthogonal to γ with regard to the inner product on gl(n): define r, ξ 1 , ξ 2 by Thenγ = ξ 1 + ξ 2 , γ, ξ 2 µ,µc,κ = 0 (4.14) and thus Without loss of generality we assume M = 0. Since γ is continuous and γ(1) = M , we can choose a := max{s ∈ [0, 1] | γ(s) = 0} such that γ(t) = 0 for all t > a. We obtain  The curve γ is piecewise differentiable by assumption, so choose a = a 1 < . . . < a n < a n+1 = 1 such that γ is continuously differentiable on (a, 1) \ {a 1 , . . . , a n }. We compute Finally, Lemma 1.7 yields We can now give a lower bound for the length of a curve (4.18) Thus, using (4.10), we find L(Z) ≥ ε and therefore Furthermore, for M µ,µc,κ < ε, we can directly compute the distance between A and for all B = A. Therefore dist geod defines a metric on GL + (n). Note also that equality in (4.15) holds if and only if .ξ 2 , and since DΦ is non-singular in a neighbourhood of 0, this equality holds (for small enough ε) if and only if for all t ∈ [0, 1]. But then γ andγ are linearly dependent, and thus γ is a parameterization of a straight line in gl(n) connecting 0 and M . Hence Φ A • γ is a reparametrization of the curve t → Φ A (tM ) = X(t), which shows that this length minimizing curve is (locally) uniquely determined. The above considerations are summarized in the following lemma: Lemma 4.4. Let A ∈ GL + (n). Then there exists ε(A) > 0 such that for all 0 < ε < ε(A): ii) for every B ∈ B ε (A) there exists a length minimizer connecting A and B, which is uniquely determined up to reparameterization; Corollary 4.5. The geodesic distance dist geod is a metric on GL + (n).

Global existence of minimizers
After these preparations, we can now prove the global existence of length minimizing curves. We will closely follow the proof of the Hopf-Rinow theorem as given in [12,Theorem 1.4.8]. Proof. Let B ∈ GL + (n) be fixed. For A ∈ GL + (n), we define r A := dist geod (A, B) and construct a length minimizer X A : [0, r A ] → GL + (n) as follows: Choose ε(A) as in Lemma 4.4 and ε with 0 < ε < ε(A), ε < dist geod (A, B). Then the continuous function attains a minimum at some M A ∈ S ε (0). Note that by this definition, Φ A (M A ) is closest to B among the geodesic sphere S ε (A). Next we define X A ∈ C ∞ ([0, r A ]; GL + (n)) by (4.25) To simplify notation, denote by L t (X A ) := L(X A [0,t] ) the length of X A up to t ∈ [0, r A ]. Then equation (3.10) yields L t (X A ) = t ε −1 M A µ,µc,κ = t. Thus we find dist geod (X A (t), A) ≤ t, and therefore (4.26) We will denote by the set of all t where equality holds in (4.26). Geometrically, I A measures for how long X A runs "in an optimal direction towards B". Since L(X A ) = L r A (X A ) = r A = dist geod (A, B), X A is a length minimizer between A and B if X A (r A ) = B, thus it remains to show that r A ∈ I A . We will first show that ε ∈ I A :  ∈ (a, b) and, according to Lemma 4.4 (4.29) in contradiction to the definition of M A . Therefore ε ∈ I A , and in particular I A is nonempty. Now assume r A / ∈ I A . It is not difficult to see that I A is closed, so lett := max I A = r A and, forÃ := X A (t), chooseε < ε(Ã), MÃ, XÃ, IÃ accordingly. We findε ∈ IÃ, and thus dist geod (XÃ(ε), B) = dist geod (Ã, B) −ε. LetX denote the piecewise smooth curvê we infer thatX is a length minimizer between A andX(t+ε). Since X A , XÃ have (the same) constant speed, X is a piecewise smooth energy minimizer in AX (t+ε) A . Then according to Proposition 2.4,X is smooth and satisfies (2.26) everywhere. Finally, because X A satisfies the differential equation with the same initial values asX, the uniqueness from Proposition 2.8 yieldŝ (4.32) and therefore in contradiction to the choice oft. Therefore r A ∈ I A , and thus dist geod (X A (r A ), B) = r A − r A = 0. Since dist geod is a metric on GL + (n) according to Corollary 4.5, we find X(r A ) = B. Since L(X A ) = r A , this concludes the proof.

Conclusion
As above, let g be a left-invariant, right-O(n)-invariant Riemannian metric on GL(n) given by  In particular, the set is non-empty for all A, B ∈ GL + (n). let alone one that is minimal with regard to the norm . µ,µc,κ . We will therefore consider some easier special cases.

The geodesic distance on GL + (1)
In the one dimensional case, we can identify GL + (1) with R + and the matrix multiplication with the usual multiplication on R. The most general inner product on R ∼ = gl(1) is given by x, y κ = κ x y, κ > 0 , (5.2) with the corresponding left-invariant Riemannian metric 3) The length and energy of a piecewise differentiable curve X ∈ C 0 ([0, 1]; R + ) are therefore It is easy to see that, in order to minimize the length over all curves connecting p, q ∈ R + , we can assume that X is strictly monotone. In this case, X is uniquely determined by p and q up to a reparameterization. We recall from Lemma 1.8 that a curve X ∈ A q p is an energy minimizer if and only if it is a length minimizer of constant speed. Since the length is invariant under reparameterization (Lemma 1.6), L is constant on the set of strictly monotone curves X ∈ A q p , and therefore any X with is an energy minimizer. To solve (5.5), we define and check g X(t) (Ẋ(t),Ẋ(t)) = κẊ (t) 2 X(t) 2 = κ p 2 exp(t ln( q p )) 2 ln( q p ) 2 p 2 exp(t ln( q p )) 2 ≡ κ(ln( q p )) 2 , (5.7) as well as Thus X is an energy minimizer, and its energy and length are given by We conclude: Proposition 5.1. The geodesic distance between p, q ∈ GL + (1) ∼ = R + is dist geod (p, q) = κ |ln( q p )| = κ |ln(q) − ln(p)| , (5.10) and a shortest geodesic connecting p and q is given by X : [0, 1] → GL + (1), X(t) = p · exp(t ln( q p )) . (5.11)

Normal matrices
The following lemma states some properties of normal matrices and their relation to the matrix exponential. Proof. i) can be shown by direct computation:

Geodesics with normal initial tangents
Let N ∈ gl(n) be a normal matrix. Then, according to Theorem 3.1, the geodesic curve X : [0, 1] → GL(n) with is the unique solution to the geodesic equation with the initial values According to Lemma 5.2, sym N and skew N commute if N is normal. But then λ 1 sym N and λ 2 skew N commute as well for all λ 1 , λ 2 ∈ R, and thus exp(λ 1 sym N + λ 2 skew N ) = exp(λ 1 sym N ) exp(λ 2 skew N ) according to (A.6). This allows us to simplify X: we find This representation of geodesics with normal initial tangents can also be found in [24,Section 8.5.1]. The length of X is and X(0) = 1, X(1) = exp(N ). In particular, X(1) is normal according to Lemma 5.2 as it is the exponential of a normal matrix.

Geodesics connecting 1 with a normal matrix
Now let A ∈ GL + (n) be normal. To find a geodesic X connecting 1 and A, we need to find M ∈ gl(n) solving But, again due to Lemma 5.2, a normal matrix A has a (generally not uniquely determined) normal logarithm, i.e. there exists a normal matrix Log A ∈ gl(n) such that exp(Log A) = A. According to (5.16), with N = Log A, the geodesic X with initial tangent Log A has the form and hence Thus for normal matrices A, the curve X : [0, 1] → GL + (n) with X(t) = exp(t Log A) is a geodesic connecting 1 and A, and (5.17) yields L(X) = Log A µ,µc,κ . We therefore obtain the upper bound dist geod (1, A) ≤ L(X) = Log A µ,µc,κ (5.19) for the distance of a normal matrix A ∈ GL + (n) to the identity 1.
Note carefully that this does not immediately imply dist geod (1, A) = Log A µ,µc,κ : While dist geod (1, A) is indeed the length of a geodesic curve connecting 1 and A, such a geodesic is generally not uniquely determined, and it is therefore possible that X is not the shortest such geodesic. However, as shown in Lemma 4.4, geodesic curves are locally unique, which immediately implies the following proposition.
In particular, for F ∈ GL + (n) with sufficiently small F − 1 , Proposition 5.3 can be applied to the positive definite symmetric (and therefore normal) matrix F T F to obtain the distance where log(F T F ) is the symmetric principal matrix logarithm of F T F on Sym + (n) [2,10]. Note, again, that it is not obvious at this point that equality (5.20) holds for all F ∈ GL + (n) since it is not immediately clear that there is no shorter geodesic curve connecting 1 and F T F .

Application to nonlinear elasticity
An example for the application of geodesic distance measures is the theory of nonlinear elasticity (and, more specifically, hyperelasticity), where the deformation of a solid body is modelled via an energy functional W which depends on the gradient F ∈ GL + (n) of the deformation (see e.g. [5,Chapter 4] for an introduction to hyperelasticity). Similarly, Mielke's work on the geodesic distance in SL(n) was primarily motivated by elasto-plastic applications [18]. Among the energy functions considered in nonlinear elasticity is the isotropic Hencky strain energy which was introduced in 1929 by Heinrich Hencky [9]. Note that, because log √ F T F is symmetric, W can be written as for arbitrary µ c > 0. As was shown by Neff et al. [22,21], the Hencky energy can be characterized as the geodesic distance (with respect to a left-GL(n)-invariant, right O(n)-invariant Riemannian metric) of the deformation gradient F to the group SO(n) of rigid rotations. The proof of this result employs the parametrization of geodesic curves given in Theorem 3.1 and the characterization of the geodesic distance stated in Corollary 4.8 as well as a recently discovered optimality result regarding the matrix logarithm [14,23].
Proposition 5.4. Let g be the left-invariant Riemannian metric on GL(n) induced by the isotropic inner product ·, · µ,µc,κ on gl(n) with µ c ≥ 0, and let F ∈ GL + (n). Then denotes the geodesic distance of F to SO(n).

Open Problems
Although the explicit parametrization of geodesic curves makes it possible to establish some lower bounds for the geodesic distance in certain special cases, there is no known general formula or algorithm to compute the distance between two elements A, B of GL + (n). However, recent results indicate that it might be possible to compute the geodesic distance for a number of additional special cases, including the (non-local) distance between arbitrary A, B ∈ SO(n) regarded as elements of GL(n) [17]. Note that although the geodesic distance on SO(n) with respect to the canonical bi-invariant metric is already well known [19], the distance discussed here takes into account the length of connecting curves which do not lie completely in SO(n). Furthermore, it might be useful to obtain some basic geometric properties of GL(n) or SO(n) with the considered metrics (e.g. to explicitly compute the curvature tensors).

A Appendix
A.1 Basic properties of the matrix exponential A proof for the following elementary properties of the matrix exponential can be found in [10]. viii) M ∈ so(n) ⇒ exp(M ) ∈ SO(n) , is called the principal logarithm on Sym + (n) and is denoted by log.
Proof. We refer to [2, Proposition 11.4.5] for a proof that exp : Sym(n) → Sym + (n) is indeed injective and infinitely differentiable. To see that its inverse is differentiable, we refer to [11,Theorem 6.6.14], where it is shown that a primary matrix function F defined through a real valued function f acting on its eigenvalues is differentiable if f is smooth on the set of eigenvalues attained on the domain of F . Since the principal logarithm on Sym + (n) is such a function defined through f = ln, and because ln is smooth on R + , i.e. on the set of eigenvalues attained on Sym + (n), it is differentiable. For further information on the matrix exponential, the matrix logarithm and matrix functions in general we refer to [10].
Proof. Choose a = a 0 < · · · < a m+1 = b such that X is continuously differentiable on has the desired properties. To see the uniqueness of ϕ, note that if both X and X •φ have constant speed, then for t ∈ [a, b] \ {a 0 , . . . , a m+1 }. Then the restrictionsφ(a) = a,φ(b) = b andφ > 0 only allow forφ ≡ 1. where Sym(n) and so(n) denote the set of symmetric and skew symmetric matrices in R n×n respectively.
Proof. We first show that for M ∈ gl(n) the matrices dev sym M, skew M and 1 are pairwise perpendicular with respect to the canonical inner product ·, · . Lemma A.5 yields dev sym M, skew M = 0 (note that dev sym M is symmetric) as well as