In this short note, we will prove some simple extensions of Obata-type theorems. Let be a Riemannian manifold. Let be the space of -tensors and we will denote the tensor product by . We will simply denote by .

**1. Condition to be a sphere **

The classical Obata’s theorem states that for a complete Riemannian manifold such that there exists a non-trivial function with , then is isometric to the standard unit sphere. It is interesting to see if there are other equations for which the existence of a non-trivial solution will ensure to be a sphere. E.g. if satisfies , then obviously also satisfies . Suppose satisfies the latter equation, is necessarily a sphere? In this section I will answer this question (affirmatively), with the assumption of a compactness condition.

Remark 1We remark that the condition for to be compact cannot be omitted in general. For example, the function on the real line satisfies . On the other hand, if satisfies such that is nontrivial, then by the classical Obata’s theorem applied to , the compactness assumption can be replaced by the completeness assumption.

Remark 2It is natural to ask if the assumption that are the only purely imaginary roots is necessary. E.g. if the only purely imaginary roots are , , can we still deduce that is a sphere (of certain radius)?

*Proof of Theorem 1:* The necessity is clear. Indeed, for , the height function when restricted to satisfies the given condition: it is known that

where is the position function, is the second fundamental form and is the unit outward normal of . Thus for , we have

From this it is easy to see that (1) is true.

Conversely suppose has a function which satisfies (1). Take any , then on each geodesic starting from parametrized by arclength, the function (when restricted to ) satisfies the linear ODE with constant coefficient

for some and (multiplicity of the root for the polynomial), and . As is bounded, it is easy to see that for all . Thus

By (3), we can deduce that . Indeed, if is an orthonormal basis which diagonalizes , then for fixed , for the geodesic emanating from such that , we have, by (3),

Since and are arbitrary, we have

We then deduce by Obata’s theorem that is the standard unit sphere.

**2. A splitting result **

In this section, I will show a simple splitting result. The proof is quite simple, and I suspect that it should be known to the experts.

*Proof:* It is easy to see that is a parallel Killing vector field. In particular, is a non-zero constant everywhere. In particular, is a smooth hypersurface. We claim that is connected and is totally geodesic. Let and let be a geodesic in with , . Then on , satisfies , and . In particular, this implies and so is connected. On the other hand, if is a geodesic in such that . Then we have . Therefore and so is totally geodesic. By Hopf-Rinow theorem, is also complete. We now claim that . Let be the one-parameter family of diffeomorphism generated by the vector field . Define by . It is easy to see that is smooth bijection, with inverse given by , so is a diffeomorphism. Since is a Killing vector field, it follows that is isometric to .

Corollary 3Let be an -dimensional connected Riemannian manifold. Let (4). Suppose . Then there is a complete connected totally geodesic submanifold of dimension such that is isometric to . The converse also holds.

*Proof:* The case for is trivial and the case is shown by Theorem 2. Suppose are independent. By the proof of Theorem 2, we have , where . By the total geodesic-ness of , the function when restricted to satisfies (4) and is non-trivial. Thus by induction, it is easy to see that the assertion is true.

Remark 3After some googling, I found that Corollary 3 has already been proved in a paper of Wu and Ye (Theorem 5.2).