Geodesic Vector Fields on a Riemannian Manifold

A unit geodesic vector field on a Riemannian manifold is a vector field whose integral curves are geodesics, or in other worlds have zero acceleration. A geodesic vector field on a Riemannian manifold is a smooth vector field with acceleration of each of its integral curves is proportional to velocity. In this paper, we show that the presence of a geodesic vector field on a Riemannian manifold influences its geometry. We find characterizations of n-spheres as well as Euclidean spaces using geodesic vector fields.


Introduction
Let (M, g) be an n-dimensional Riemannian manifold. We call a smooth vector field ξ on M geodesic vector field if where ∇ is the covariant derivative operator with respect to the Riemannian connection on (M, g) and ρ : M → R is a smooth function called the potential function of the geodesic vector field ξ. If the potential function ρ = 0, then ξ is called a unit geodesic vector field (as in this case the integral curves of ξ are geodesics). By a non-trivial geodesic vector field, we mean nonzero geodesic vector field for which the potential function ρ = 0. Physically, a geodesic vector field has integral curves with an acceleration vector always proportional to the velocity vector. These fields are connected with generalized Fermi coordinates [1]. Geodesic vector fields naturally appear in many situations as seen in the following examples: 1.

2.
Consider unit hypersphere S n in the Euclidean space (R n+1 , , ). Then, the restriction of coordinate vector field ∂ ∂u 1 on R n+1 to S n can be expressed as where ρ = ∂ ∂u 1 , N , N being unit normal to S n and ξ is vector field on S n , which is the tangential component of ∂ ∂u 1 . Then it is easy to see that on S n , we have ∇ ξ ξ = ρξ, that is, ξ is a geodesic vector field on S n . 3.
Concircular vector fields on Riemannian manifolds have been introduced by A. Fialkow (cf. [2,3]). A vector field ξ on a Riemannian manifold (M, g) is said to be a concircular vector field if ∇ X ξ = ρX for any smooth vector field X on M, where ρ is a smooth function on M. Thus, a concircular vector field ξ satisfies ∇ ξ ξ = ρξ, that is, a concircular vector field ξ is a geodesic vector field. It is well known that concircular vector fields play a vital role in the theory of projective and conformal transformations. Moreover, concircular vector fields have applications in general relativity, as for instance trajectories of time-like concircular fields in the de Sitter space determine the world lines of receding or colliding galaxies satisfying the Weyl hypothesis (cf. [4]). Therefore, we could expect that geodesic vector fields also have the scope of applications in general relativity. For example, global questions about the existence of these vector fields were studied in [5][6][7][8][9][10]. 4.
Another interesting example comes from Yamabe solitons (cf. [11,12]). Let (M, g, ξ, λ) be an n-dimensional Yamabe soliton. Then the soliton field ξ satisfies where £ ξ g is the Lie-derivative of metric g, S is the scalar curvature and λ is a constant. If the soliton field ξ is a gradient of a smooth function, then (M, g, ξ, λ) is called a gradient Yamabe soliton. On gradient Yamabe soliton the soliton field satisfies ∇ ξ ξ = (S − λ)ξ, that is, ξ is a geodesic field with potential function ρ = S − λ.

5.
Recall that an Eikonal equation is a nonlinear partial differential equation where on a non-compact Riemannian manifold (M, g), which is encountered in problems of wave propagation, where f is a positive function (cf. [13,14]). A straight forward observation shows that, above equation gives ∇ ∇u ∇u = − 1 f 3 ∇ f , which on choosing u = 1 f , gives ∇ ∇u ∇u = u∇u, that is, an Eikonal equation gives a non-trivial geodesic vector field ∇u with potential function u. Note that Eikonal equations are also used in tumor invasion margin on Riemannian manifolds of brain fibers (cf. [15]).
Geodesic vector fields first time appeared in [12] as generalization of unit geodesic vector fields, where they are used for finding conditions under which a Yamabe soliton is trivial. As observed through above examples, geodesic vector fields have widespread appearance as compared to Killing vector fields and conformal vector fields, which suggests that they may have a role not only in the geometry of a Riemannian manifolds, but also in theory of relativity as well as medical imaging via the Eikonal equation. In this paper, we concentrate on the first two examples of geodesic vector fields mentioned above. Example 1 shows that the Euclidean space (R n , , ) possesses a geodesic vector field, naturally raises a question: "Under what conditions does a Riemannian manifold have a geodesic vector field necessarily isometric to the Euclidean space?" A similar question is raised through Example 2 mentioned above. In this paper, we address these questions and find characterizations of the n-sphere S n (c) as well of the Euclidean space (R n , , ) using geodesic vector fields (cf. Theorems 1 and 2).

Preliminaries
Let ξ be a geodesic vector field on an n-dimensional Riemannian manifold (M, g) with potential function ρ. We denote by α the smooth 1-form dual to ξ. Then we have where ∇ is the covariant derivative operator with respect the Riemannian connection on (M, g) and X(M) is the Lie algebra of smooth vector fields on M. Note that the Lie derivative £ ξ g is symmetric, while the smooth 2-form dα is skew-symmetric, which give a symmetric operator B and a skew-symmetric operator ψ on M defined by Then using Equations (2) and (3), we conclude Using the defining Equation (1) of geodesic vector field in Equation (4), we get The curvature tensor field R and the Ricci tensor Ric of the Riemannian manifold (M, g), are given by and where {e 1 , .., e n } is a local orthonormal frame on M. The Ricci operator Q of the Riemannian manifold (M, g) is a symmetric operator defined by The scalar curvature S of the Riemannian manifold is defined by S = TrQ the trace of the Ricci operator Q. The gradient ∇S of the scalar curvature satisfies (cf, [30]) where the covariant derivative Choosing Y = Z = ξ in Equation (6) and using Equations (1) and (4), we conclude Taking X = e i in above equation and the inner product with e i , on summing the resulting equation over an orthonormal frame {e 1 , .., e n }, we get where f = TrB the trace of the symmetric operator B, we have used Trψ = 0 (ψ being skew-symmetric) and the fact that Bψ + ψB is a skew-symmetric operator and We associate one more smooth function h : M → R on a Riemannian manifold (M, g) to geodesic vector field ξ, defined by Then, using Equation (4), we get the following expression for the gradient ∇h of the smooth function h, Note that for a smooth function F : M → R on a Riemannian manifold (M, g), the Hessian operator A F and the Laplacian ∆F are defined by The Hessian Hess(F) is defined by

A Characterization of Euclidean Spaces
In this section, we use a non-trivial geodesic vector field on a connected Riemannian manifold to find a characterization of the Euclidean spaces. We have seen through Example-1 in the introduction that the Euclidean space (R n , , ) admits a geodesic vector field ξ with potential function ρ a constant. Recall that a geodesic vector field ξ with potential function ρ is said to be a non-trivial geodesic vector field if ξ is nonzero and ρ = 0. Theorem 1. Let (M, g) be an n-dimensional complete and connected Riemannian manifold. The following two statements are equivalent: 1. There exists a non-trivial geodesic vector field ξ with potential function ρ with the properties that Tr£ ξ g is constant along the integral curves of ξ and Ricci curvature Ric(ξ, ξ) satisfies 2. (M, g) is isometric to Euclidean space (R n , , ).
Proof. Suppose that ξ is a non-trivial geodesic vector field on the connected Riemannian manifold (M, g), such that ξ( f ) = 0, where f = 1 2 Tr£ ξ g = TrB and the Ricci curvature Ric (ξ, ξ) satisfies Now, as dα(X, Y) = 2g (ψX, Y), we get 1 4 dα 2 = ψ 2 and the above inequality takes the form Using Equation (10) with ξ( f ) = 0, we get that is, Now, using the inequality (15) in the above equation, we conclude However, by Schwartz's inequality, we have B 2 ≥ 1 n f 2 and the equality holds if and only if B = f n I. Thus, inequality (17), implies B = f n I.
Using Equations (5) and (18), we conclude and taking the inner product with ξ in the above equation and noting that ψ is skew-symmetric, we get As ξ is non-trivial, ξ = 0 and consequently, on connected M above two equations give Combining Equations (12), (18) and (19), we conclude ∇h = f n ξ, which on using Equation (4), gives

Thus, the Hessian Hess(h) is given by
Now, using the facts that Hess(h) is symmetric and the operator ψ is skew-symmetric in above equation, we conclude that is, Taking X = ξ in above equation and using Equation (19), we get ξ 2 ∇ f = ξ( f )ξ = 0 by the assumption in the statement. Since, ξ is non-trivial geodesic vector field, the equation ξ 2 ∇ f = 0 on connected M, implies ∇ f = 0, that is, f is a constant. Note that the constant f has to be a nonzero constant, for if f = 0, then Equation (19) would imply ρ = 0, which is a contradiction to the fact that ξ is a non-trivial geodesic vector field. Using this fact that f is a nonzero constant in Equation (21), we conclude ψ = 0. Hence, Equation (20), takes the form where c is a nonzero constant. Finally, we observe that the smooth function h is not a constant, for if not, then the Equation (12), would imply ξ = 0, a contradiction to the fact that ξ is a non-trivial geodesic vector field. Hence, Equation (22) on a complete and connected Riemannian manifold (M, g) implies that (M, g) is isometric to the Euclidean space (R n , , ) (cf. [31], Theorem 1, p. 778, [14]). Conversely, on the Euclidean space (R n , , ), we have the position vector field which satisfies ∇ X ξ = X, X ∈ X(R n ), where ∇ is the covariant derivative with respect to the Euclidean connection. Then, it follows that ξ is the non-trivial geodesic vector field with potential function ρ = 1 and corresponding operators B = I and ψ = 0. Thus, f = TrB = n is a constant and Ric (ξ, ξ) = 0, that is, we get which meet the requirements in the statement.

A Characterization of n-Spheres
In this section, we use non-trivial geodesic vector field on a compact and connected Riemannian manifold to find a characterization of a n-sphere S n (c). Indeed we prove the following: 2. (M, g) is isometric to n-sphere S n (c).
Proof. Let ξ be a non-trivial geodesic vector field on an n-dimensional compact and connected Riemannian manifold (M, g) of constant scalar curvature, with potential function ρ satisfying the condition in the statement. Since, f = TrB = 1 2 Tr£ ξ g and ψ 2 = 1 4 dα 2 , the condition in the statement reads Using Equation (4), we get divξ = f and consequently, div f ξ = ξ( f ) + f 2 and divρξ = ξ(ρ) + ρ f .
Integrating these equations, we conclude Now, integrating Equation (10) and using Equation (24), we get Next, we use the inequality (23) in the above equation, to conclude However, by Schwartz's inequality, we have B 2 ≥ 1 n f 2 , that is, and combining this inequality with inequality (26), we conclude Thus, using Schwartz's inequality, we get B 2 = 1 n f 2 and this equality holds if and only if B = 1 n f . Moreover, Equation (25) implies Using B = f n I, and following the proof of Theorem 1, through Equations (18)- (21), we conclude Taking X = ξ in above equation and using ψξ = 0, we have ξ 2 ∇ f = ξ( f )ξ, which on taking the inner product with ∇ f , gives Using a local orthonormal frame {e 1 , .., e n } on M, Equation (28), gives and summing these equations, leads to Thus, using Equation (29), we conclude f 2 ψ 2 = 0. Note that if f = 0, then Equation (19), gives ρ = 0, which is contrary to our assumption that ξ is non-trivial geodesic vector field. Hence, on connected M equation f 2 ψ 2 = 0 implies that ψ = 0. Now, Equation (5) transforms to which on using Equation (6), gives the following expression for the curvature tensor Using Equation (33) in above equation, we get Note that Equation (32), gives Inserting this equation in Equation (37), leads to In this equation, we use the facts that Ric > 0 and the Schwartz's inequality A f Taking the covariant derivative in the first equation of Equation (38) with respect to X ∈ X(M) and using Equation (30), we get where c is a positive constant given by S = n(n − 1)c. Note that, we have ruled out above that f can be a constant. Hence, the non-constant function f satisfies the Obata's differential Equation (39) (cf. [26]) and consequently, the Riemannian manifold (M, g) is isometric to the sphere S n (c). Conversely, if (M, g) is isometric to S n (c), then the Ricci curvature for any smooth vector field X on S n (c) is given by Ric(X, X) = (n − 1)c X 2 . We treat S n (c) as hypersurface of the Euclidean space R n+1 , , with unit normal vector field N and the shape operator A = − √ cI. Now, choosing a nonzero constant vector field w ∈ X(R n+1 ), we express its restriction to the sphere S n (c) as w = ξ + sN, where ξ is tangential component of w to S n (c) and s = w, N is the smooth function on S n (c). Taking covariant derivative with respect to X ∈ X(S n (c)) of the equation w = ξ + sN and using Gauss and Weingarten formulas for the hypersurface, we get Equating tangential and normal components in the above equation, we get The first equation in Equation (40) gives ∇ ξ ξ = ρξ, where ρ = − √ cs. This proves that ξ is a geodesic vector field with potential function ρ. Suppose ρ = 0, this will mean s = 0 and consequently, the second equation in Equation (40) will imply that ξ = 0. Thus, w = 0 on S n (c), but as w is a constant vector field, we get w = 0 on R n+1 , contrary to our assumption that w is a nonzero constant vector field. Hence, ρ = 0. Similarly, we can show that ξ is a nonzero vector field. Hence, ξ is a non-trivial geodesic vector field on S n (c). Next, by second equation in the Equation (40), we have c ξ 2 = ∇s 2 , and that Also, by the first equation in (40), for the geodesic vector field ξ, the operators B and ψ are B = − √ csI and ψ = 0, and that f = TrB = −n √ cs. Moreover, using Equation (40), we find divξ = −n √ cs and ∆s = −ncs. Thus, we get S n (c) ∇s 2 = nc S n (c) Finally, using Equations (41) and (42), we conclude S n (c) Ric(ξ, ξ) = n − 1 n S n (c) which is the equation in (23), that is, all the requirements in the statement are met.