A new geometric structure on tangent bundles

For a Riemannian manifold $(N,g)$, we construct a scalar flat metric $G$ in the tangent bundle $TN$. It is locally conformally flat if and only if either, $N$ is a 2-dimensional manifold or, $(N,g)$ is a real space form. It is also shown that $G$ is locally symmetric if and only if $g$ is locally symmetric. We then study submanifolds in $TN$ and, in particular, find the conditions for a curve to be geodesic. The conditions for a Lagrangian graph to be minimal or Hamiltonian minimal in the tangent bundle $T{\mathbb R}^n$ of the Euclidean real space ${\mathbb R}^n$ are studied. Finally, using the cross product in ${\mathbb R}^3$ we show that the space of oriented lines in ${\mathbb R}^3$ can be minimally isometrically embedded in $T{\mathbb R}^3$.


Introduction
The geometry of the tangent bundle T N of a Riemannian manifold (N, g) has been a topic of great interest for the last 60 years. In the celebrated article [12], Sasaki used the Levi-Civita connection of g to split the tangent bundle T T N of T N into a horizontal and a vertical part, constructing the first geometric structure of T N. . Namely, one can obtain a splitting T T N = HN ⊕ V N, where the subbundles HN and V N of T T N are both isomorphic to the tangent bundle T N -for more details Date: 14th June 2018. 1 see section 2. ForX ∈ T T N, we writeX ≃ (ΠX, KX), where ΠX ∈ HN and KX ∈ V N. Sasaki defined the following metric on T N [8]: (X,Ȳ ) → g(ΠX, ΠȲ ) + g(KX, KȲ ), The Sasaki metric is "rigid" in the following sense: it is scalar flat if and only if g is flat [9]. In the years since, several new geometries on the tangent bundle T N have been constructed using the splitting of T T N -see for example [10] and [13]. When the base manifold N admits additional structure one can use it to define other geometries in the tangent bundle. H. Anciaux and R. Pascal constructed in [3] a canonical pseudo-Riemannian metric in T N derived from a Kähler structure on N. One example, the canonical neutral metric in T S 2 defined by the standard Kähler structure of the round 2-sphere S 2 , has been used to study classical differential geometry in R 3 -see [1], [6] and [7]. Using the Riemannian metric g one can define a canonical symplectic structure Ω in T N. This can be achieved by using the musical isomorphism between the tangent bundle and the cotangent bundle. In this article we use the existence of an almost paracomplex structure J on T N compatible with Ω to construct a neutral metric G on T N. If the base manifold N admits a Kähler structure, the neutral metric G and the pseudo-Riemannian metric, derived from the Kähler structure, are isometric -see Propostion 5. In other words, the neutral metric G is a natural extension of the Kähler metric constructed by H. Anciaux and R. Pascal in [3] to the case where the base manifold does not admit a Kähler structure. The purpose of this article is to study the geometric properties and submanifolds of the neutral metric G. We first prove the following: Theorem 1. The neutral metric G has the following properties: (1) G is scalar flat, (2) G is Einstein if and only if g is Ricci flat, (3) G is locally conformally flat if and only if either n = 2 or, g is of constant sectional curvature, (4) G is locally symmetric if and only if g is locally symmetric.
We then focus our attention on submanifolds. In particular, the geodesics of G are characterized by: Theorem 2. A curve γ(t) = (x(t), V (t)) in T N is a geodesic with respect to the metric G if and only if the curve x is a geodesic on N and V is a Jacobi field along x.
It was shown in [2], that the existence of a minimal Lagrangian graph in T N, where (N, g) is a 2-dimensional Riemannian manifold, implies that g is flat. A generalization of this result is given by the following Theorem: Theorem 3. If T N contains a Lagrangian graph with parallel mean curvature then the neutral metric G is Ricci flat.
To continue with submanifolds, we need to introduce some more terminology. A vector field X in a symplectic manifold (N, ω), is called a Hamiltonian field if ω(X, .) = dh, where h is a smooth function on N. A Lagrangian immersion will be called Hamiltonian minimal if the variations of its volume along all Hamiltonian vector fields are zero. Let Σ be a Lagrangian submanifold in a (para-) Kähler manifold (N, g, ω, j), and H be its mean curvature. The first variation formula shows that Σ is Hamiltonian minimal if and only if the tangential vector field jH is divergence free [14]. For Lagrangian graphs in T R n we prove the following: Theorem 4. Suppose that u is a C 4 -smooth function in an open set of R n and let f be the corresponding Lagrangian graph f : R n → T R n : p → (p, Du(p)) in T R n . Then the following two statements hold true: (1) If u is functionally related of second order then the induced metric f * G is flat.
where c 0 is a positive real constant. Furthermore, f is totally geodesic if and only if u is of the following form: where a ij , b i and c are all real constants and det a ij > 0. In particular, every totally geodesic Lagrangian graph is flat.
As an example of Theorem 4, we focus our attention to the graph of a vector field in R n of spherical symmetry, called source fields (see Definition 3). Such graphs are Lagrangian submanifolds in T R n .
Then the following two statements hold: (1) f is minimal if and only if the field intensity H is given by where c 0 , c 1 are constants with c 0 > 0. Furthermore, f is totally geodesic if and only if c 1 = 0. (

2) f is Hamiltonian minimal (but non minimal) if and only if the field intensity H is given by
It is well known that the space L(R 3 ) of oriented lines in R 3 is identified with the tangent bundle T S 2 . Guilfoyle and Klingenberg in [6] and M. Salvai in [11], studied the geometry of (L(R 3 ), G, J) derived by the standard Kähler structure (T S 2 , g, J).
Using this identification we show the following: There exists a minimal isometric embedding of (L(R 3 ), G) in (T R 3 , G).
The paper is organized as follows. In the next section the new geometry is introduced in the context of almost Kähler and almost para-Kähler structures. Theorem 1 is proven in section 3, while proofs of Theorems 2, 3, 4 and 5 are contained in sections 4.1, 4.2, 4.3, and 4.4, respectively. Theorem 6 is proven in the final section.

Almost (para-) Kähler structures on TN
Let N be an n-dimensional differentiable manifold and π : T N → N be the canonical projection from the tangent bundle T N to N. We define the vertical bundle V N as the subbundle Ker(dπ) of T T N. If N is equipped with an affine connection D, then we may define the horizontal bundle HN of T T N as follows: IfX is a tangent vector of T N at (p 0 , V 0 ), there exists a curve a(t) = (p(t), V (t)) ⊂ T N such that a(0) = (p 0 , V 0 ) and a ′ (0) =X. Define the connection map (see [5] and [9], for further details) K : T T N → T N by KX = D p ′ (0) V (0) ifX = V N (i.e., p ′ (0) = 0) and ifX ∈ V N (in this case,X is said to be a vertical vector field) then KX = V ′ (0). The horizontal bundle HN is simply Ker(K) and therefore we obtain the direct sum: Given a vector field X on (N, D) there exist unique vector fields In addition, if X, Y are vector fields of N, we have at (p, V ) ∈ T N: where R denotes the curvature of D.
If g is a Riemannian metric on N, we identify the cotangent bundle T * N with T N by the following bundle isomorphism: g(p, X) = g p (X, .) for any X ∈ T p N. Using the canonical projection π * : T * N → N, we define the Liouville form ξ ∈ Ω 1 (T * N) by: The derivative of the Liouville form defines a canonical symplectic structure, Ω * := −dξ, on T * N and using the isomorphism g we define a symplectic structure Ω on T N by Ω = g * Ω * . The symplectic structure Ω is given by The Sasaki metrics G 0 and G 2 are very well known and have been studied extensively by several authors -see for example [4,8,13]. In this article we fill the gap by studying the geometry of (T N, G 1 ). From now on and throughout this article, we simply write G for G 1 .

Curvature of the neutral metric G
Consider the neutral metric G constructed in Section 2. We now study the main geometric properties of (T N, G).
Denote the Levi-Civita connection of G by ∇. For a vector field X in N we use Proposition 1, to consider the unique vector fields X h and Using the Koszul formula: one may obtain the following relations We now have at (p, V ) ∈ T N, and using the first Bianchi identity we finally get, Similar calculations give, Using (5) and (6) we obtain where X = ΠX h and Y = ΠY h . Putting all together we find that where Rm is the Riemann curvature tensor of g and Proof. Using the second Bianchi identity, we have at the point (p, V ) ∈ T N : The proposition follows by using the fact that We are now in position to calculate the Ricci tensor: Proposition 4. The Ricci tensor Ric of the metric G is given by where Ric denotes the Ricci tensor of g.
Proof. Following a similar method as in the proof of the main theorem of [3], we consider the orthonormal frame (e 1 , . . . , e n ) of (N, g). Define the following frame (ē 1 , . . .ē n ,ē n+1 , . . .ē 2n ) of T N to be the unique vector fields such that and this completes the Proposition.
Proof of Theorem 1. 1. For the first part consider, as before, the frame (ē 1 , . . .ē n ,ē n+1 , . . .ē 2n ) of T N. The scalar curvatureS of the metric G is 2. The second part follows directly from the Proposition 4. 3. We now prove the third part. Let W be the Weyl tensor of G. Using Proposition 5 of [3] we have Assume that G is locally conformally flat and n ≥ 3. We then have Let X, Y, Z, W be vector fields on N with corresponding unique vector fields and thus, (10) becomes, We now have, On the other hand, for any vector fields X, Y on N. Thus, there exists a smooth function λ on N such that Ric(X, X) = λ|X| 2 , which shows that g is Einstein. Since n ≥ 3, the function λ must be constant. Let P be the plane spanned by {e 1 , e 2 }. Then the sectional curvature which is constant. Assume the converse, that is, n ≥ 3 and g is of constant sectional curvature K. Thus, where R denotes the scalar curvature. Also g is locally symmetric and therefore from Proposition 3 we have Hence, Rm(X h , Y h , Z h , W h ) = 0 and using (9) we have, Since g is of constant sectional curvature K, we have and therefore, using (11) we have Similarly, we prove that all coefficients of the Weyl tensor vanish and that W = 0. Thus, for n ≥ 3, the metric G is locally conformally flat if and only if g is of constant sectional curvature. For n = 2, the Riemann curvature tensor is given by where K is the Gauss curvature of g. Hence, following a similar argument as before, one can prove that for every 2-manifold (N, g) the neutral metric G of T N is locally conformally flat. 4. We now proceed with the last part of the proof. Assume first that g is locally symmetric. Then for any vector fields ξ, X, Y, Z on N we have, by definition, Using (12), a brief computation shows We thus obtain the followinḡ Applying the relations (4) and (7) we get We now use all relations above and together with (13) finally obtain, Similar arguments shows that this relation holds: showing that ∇R = 0, which means G is locally symmetric. Conversely, assume that G is locally symmetric. Then the following holds true: which means that g is locally symmetric, completing the proof of the Theorem.
Suppose that the manifold N is equipped with a Kähler structure (j, g). An almost complex structure J on T N can be defined by JX = (jΠX, jKX).
It has been proved that J is integrable and one can check easily that is compatible with G, that is, G(J., J.) = G(., .). It can be easily proved that J is also parallel with respect to ∇. Namely, The complex structure J is compatible with Ω and together with the metric G given by G = Ω(J., .) defines another Kähler structure (G, Ω, J) on T N which it has been introduced by H. Anciaux and R. Pascal in [3]. In particular, The following Proposition shows that the neutral metric G is an extension of G for the non-Kähler structures.
Proof. Let N be a smooth manifold equipped with a Kähler structure (j, g). Let G and G be the Kähler metrics defined as above and define the following diffeomorphism: IfX ∈ T (p,V ) T N then Πdf (X) = ΠX and using the fact that j is parallel, we have that Kdf (X) = −jKX. Thus, which shows that f is an isometry.

Submanifold theory
We now investigate the submanifold theory of (T N, G). In particular, we will study geodesics and the Lagrangian graphs.

4.2.
Graph submanifolds. Let (N, g) be a n-dimensional Riemannian manifold and U be an open subset of N. Considering a vector field V in N, we obtain a n-dimensional submanifold V ⊂ T N which is a section of the canonical bundle π : T N → N. Such submanifolds are immersed as graphs, that is, V = f (N), where f (p) = (p, V (p)). The following proposition gives a relation between the null points of the graph V with the critical points of the length function of an integral curve of V . Proposition 6. Let V be a vector field of N and V be the corresponding graph. Then the following two statements hold true: (1) If an integral curve of V is a geodesic in (N, g) then V admits a null curve.
(2) If V admits a closed integral curve then V must contain a null point.
Proof. Let p = p(t) be an integral curve of V , that is, V (t) := V (p(t)) = p ′ (t). The corresponding curve in T N is given by Then f ′ (t) = (p ′ (t), D p ′ (t) V (t)) and thus, (1) When the curve p = p(t) is a geodesic then D p ′ (t) p ′ (t) must vanish and thus G(f ′ (t), f ′ (t)) = 0 for any t. Therefore f (t) is a null curve.
(2) Assuming that the integral curve p(t) is closed, there exists t 0 ∈ S 1 such that, which means that f is null at the point t 0 .
We now study Lagrangian graphs in T N. We need first to recall the definition of a Lagrangian submanifold: For Lagrangian graphs we have the following: We now prove our third result.
Proof of Theorem 3. Let g be the Riemannian metric in N and V be the submanifold of T N obtained by the image of the graph: where V is a vector field defined on the open subset of N. The fact that f is Lagrangian implies that the almost paracomplex structure J is a bundle isomorphism between the tangent bundle T V and the normal bundle NV. We then consider the Maslov form η on V defined by, where H is the mean curvature vector of f . The Lagrangian condition implies the following relation: where Ric denotes the Ricci tensor of G. Assuming that H is parallel, the Maslov form is closed and therefore, Ric(JX,Ȳ ) = 0, for every tangential vector fieldsX,Ȳ . If Ric denotes the Ricci tensor of g, the Proposition 4, gives Ric(X,Ȳ ) = 2Ric(ΠX, ΠȲ ), If X, Y are vector fields in U, the fact that f is a graph, implies that Πdf (X) = X and Πdf (Y ) = Y . On the other hand, using the definition J, we have J(df (X)) = (Πdf (X), −Kdf (X)).  Proof. Suppose that g is non-flat. Using Theorem 3, the Lagrangian surface Σ can't be the graph of a smooth function on N. Following a similar argument as the proof of Proposition 2.1 of [?], Σ can be parametrized by: where j denotes the canonical complex structure on N defined as a rotation on T N about π/2 and γ = γ(s) is a curve in N. The mean curvature H of f is where k denotes the curvature of γ. Obviously, we have that ∇ ∂t H = 0 and which shows that Σ has parallel mean curvature is equivalent to the fact that γ is a geodesic.

4.3.
Lagrangian graphs in the Euclidean space. In this subsection we study Lagrangian graphs in T R n .

Definition 2.
A smooth function u on R n is said to be functionally related of second order if for every two pairs (i 1 , i 2 ) and (j 1 , j 2 ) there exists a function F on Example: The following functions in R n are functionally related of second order: u(x 1 , . . . , x n ) := f (a 1 x 1 + . . . + a n x n ) and v(x 1 , . . . , Consider the n-dimensional Euclidean space (R n , ds 2 ), where ds 2 is the usual inner product and let G be the neutral metric in T R n derived by ds 2 . For Lagrangian graphs in T R n we prove the Theorem 4.
Proof of Theorem 4. Let (x 1 , . . . , x n ) be the standard Cartesian coordinates of R n and let f (x 1 , . . . , x n ) = (x 1 , . . . , x n , u x 1 , . . . , u xn ), be the local expression of the Lagrangian submanifold. 1. We then have, The first fundamental form has coefficients: whereB denotes the second fundamental form of f . The Levi-Civita connection ∇ of g is: The equation (15) gives Assuming now that u is functionally related of second order implies for any indices i, j, k, l, s, t, and this completes the first statement of the Theorem.
2. Let GL(n, R) be the space of all invertible n × n real matrices and let g : R n → GL(n, R) : (x 1 , . . . , x n ) → (g ij (x 1 , . . . , x n )) be a smooth mapping. Using Jacobi's formula, where Adj denotes the adjoint matrix of h, then one can prove the following: Lemma 1. Let g : R n → GL(n, R) be a smooth mapping. If g ij and g ij are the entries of g and g −1 , respectively, then Using the expression (16), the second fundamental form becomes The mean curvature vector H is We then have from Lemma 1, Using (18), we get G(JH, f s ) = ∂ xs log | det Hess(u)| −2 , and thus, Denote the divergence with respect to the induced metric g by div. Then, where ∆ denotes the Laplacian with respect to g. Therefore, f is Hamiltonian minimal if and only if ∆ log | det Hess(u)| = 0.
3. The minimal condition follows easily from (18). Assuming that f is totally geodesic, the relation (17) gives for every indices i, j, k and thus u must satisfy (2). Conversely, assume that u is defined by (2). Then all third derivatives vanish and obviously the second fundamental form B is identically zero. The fact that totally geodesic Lagrangian graphs are flat comes from part (1) of the Theorem.
As an application of Theorem 4, we give the following corollary: Let Ω be an embedded ball in R n such that ∂Ω is an embedded sphere. Suppose that f is a Hamiltonian minimal graph in Ω such that the induced metric is Riemannian. If f is minimal at ∂Ω then it is minimal in Ω.
Proof. Let g be the induced metric of G in Ω through f and let η be the outward unit vector field normal to ∂Ω with respect to g. Since f is a Lagrangian graph, there exists a smooth function u : Ω → R such that f (p) = (p, Du(p)), p ∈ Ω. Therefore, g = 2Hess(u) and let h = det Hess(u). The immersion f is minimal at ∂Ω and thus the Theorem 4 tells us that h is constant in ∂Ω. Using Stokes Theorem and the fact that ∂Ω is a sphere we have, where div and ∇ denote the divergence and the gradient of g. This, implies where ∆ is the Laplacian of g.
On the other hand, f is Hamiltonian minimal. Then, Theorem 4 says that log h must be harmonic. In other words, That means, g(∇h, ∇h) = h∆h, and by integrating over Ω, we then have Using (19) and (20) we have Ω g(∇h, ∇h) = 0, and since g is Riemannian it follows that ∇h(p) = 0, for every p in Ω and the Corollary is completed.

4.4.
Source fields in R n . As an application of Theorem 4, we explore the geometry of source vector fields in R n .

Definition 3.
A vector field V in R n − {0} is said to be an SO(n) invariant source field (or simply a source field) if it is given by where R is the distance to the origin (the source). The function H is called the field intensity.
We now study submanifolds in T R n that are graphs of source fields in R n . If V is the source field, the corresponding graph in R n will be denoted by V. Proof. Observe that if u ′ (R) = H(R), then and the proposition follows.
We now prove our next result: Proof of Theorem 5. Let H be the field intensity of a source field V . Then It is not hard to see that there exists a smooth function u such that V = ∇u, which means that u x i = h(R)x i . The induced metric g = f * G has coefficients Then, where I is the n × n diagonal matrix and, We where λ i are the eigenvalues of −A, which are λ 1 = . . . = λ n−1 = 0 and λ = −R 2 . Thus, where h ′ denotes differentiation with respect to R. In terms of the field intensity, we have detHess(u) = H n−1 H ′ R n−1 . The nondegenerecy of g = f * G implies that the function H monotone.
1. Using Theorem 4, the immersion f (p) = (p, ∇u(p)) is minimal if and only if detHess(u) = c 0 , where c 0 is a positive constant. This is equivalent to solving the following differential equation: Solving this ODE, we get, where c 1 is a real constant and thus the field intensity H is H(R) = Rh(R) = (c 0 R n + c 1 ) 1/n .
Suppose now that c 1 = 0. Then h(R) = c 1/n 0 and using (21) we have that u x i x j = c 1/n 0 δ ij . Hence, one can see easily that u(x 1 , . . . , x n ) = c 1/n 0 2 (x 2 1 + . . . + x 2 n ) + d, where d is a real constant. Using Theorem 4, it follows that the graph f is totally geodesic. If, conversely, f is totally geodesic, the function u must satisfy (2). Since ∇u = h(R)(x 1 , . . . , x n ), we have On the other hand, taking the derivative of (2) with respect to x k we have Therefore, b k = 0 and a ik = 0 for any i = k. Thus, h(R) = a kk for any k, which implies that a kk = a for some constant a. Then h(R) = a and it can be obtained by using (22) by setting c 1 = 0.

Special isometric embeddings
It is well known that the space L(R 3 ) of oriented lines in the Euclidean 3-space R 3 is identified with the T S 2 , where S 2 denotes the round 2-sphere. Consider the Kähler metric G of L(R 3 ) derived from the standard Kähler structure endowed on S 2 . We then prove: Proof of Theorem 6. Consider the round 2-sphere S 2 and let f : T S 2 → T R 3 : (p, V ) → (p, −p × V ) be the embedding, where × is the cross product in R 3 . For X ∈ T (p,V ) T S 2 , the derivative df (X) is given by The metric G in T S 2 is G (p,V ) (X, Y ) = g(KX, p × ΠY ) − g(ΠX, p × KY ).
For X, Y ∈ T (p,V ) T S 2 , we have which shows that f * G = G and thus f is an isometric embedding.
We now show that f is minimal. Denote by ∇, ∇ the Levi-Civita connections of (R 3 , ., . ) and (T R 3 , G), respectively and also denote respectively by D, D the Levi-Civita connections of (S 2 , g) and (T S 2 , G).
Suppose that (p, V ) ∈ T S 2 and |V | = 0. Consider the following orthogonal basis of T (p,V ) T S 2 : Thus,