[Updated on 5/10/2011: John Ma discovered that the result (re)discovered by him was proved by Lichnerowicz long time ago. ]

This is a sequel to the previous post on a condition on a Riemmannian manifold to be isometric to a sphere. In that post, we proved that is a sphere if there is a nontrivial function such that

It is natural to ask if we can weaken the assumption to the existence of a nontrivial such that

As it turns out, if we impose further condition on , then it is true. More precisely, we can prove that

Theorem 1 (Obata)For a compact Einstein manifold with positive scalar curvature , if there is a nonconstant function on such that

then is isometric to the standard unit sphere.

Recall that is Einstein if its Ricci curvature for some constant , in this case we can let and then the scalar curvature is .

Actually Theorem 1 is the corollary of the following

- For an orientable compact , for two tensors of the same type, we can define at each point , we can then define
If is non-orientable, we then take its orientable double cover and lifts to (with the same notations ), we then define (naturally all tensors, and in particular and etc, can be lifted to . )

- The tensor can be raised to a tensor and thus can be treated as a linear map , we denote the later as . If is Einstein, i.e. , then is just .
- We define as the differential operator which takes -tensors to -tensors by
for . By Stokes theorem, and are dual to each other:

for -tensor and -tensor . We define , in other words,

Note that in this notations, is non-negative definite. (See Lemma 3 (2))

With these notations in mind, we have

Lemma 3Suppose , we have

*Proof:* The first two are easy. For (3), consider

For (4), . The last formula is the consequence of the above.

As a corollary, as and , we have

This is the eigenvalue estimate of Theorem 2. In the case where is Einstein with , the condition is reduced to

Also, note that the equality holds if and only if , thus by Obata’s theorem, is isometric to a standard sphere of radius in the Euclidean space if is orientable. Actually can’t be non-orientable, otherwise, since on , we also have on , and thus both are isometric to a standard sphere of the same radius, which is impossible (as one is orientable and the other is not, note that there is no orientation assumption in Obata’s theorem).

As remarked by John Ma, we actually can weaken the assumption of being Einstein by imposing some conditions on its curvature in Theorem 1 [Updated: this result was discovered by Lichnerowicz and rediscovered by John Ma]:

*Proof:* Indeed, the above proof shows that

Since the equality is attained, and we can apply Obata’s theorem.