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 1 We 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 2 It 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 , where .
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.
Theorem 2 Suppose is a connected complete Riemannian manifold such that there exists a nontrivial function on satisfying
Then there is a totally geodesic connected complete hypersurface such that is isometric to . The converse also holds.
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 3 Let 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 3 After some googling, I found that Corollary 3 has already been proved in a paper of Wu and Ye (Theorem 5.2).