In this note, we will give a generalization of Hsiung-Minkowski formulas for hypersurfaces in space forms. This will generalize the results in some of our previous posts. For example, as a special case of Theorem 6, we have the following result:

The definition of will be given in the next section. We remark that the classical Hsiung-Minkowski formulas [Hsiung] can be recovered by putting in the above equation. By choosing suitable in Theorem 1, we can obtain the following corollary:

Corollary 2(Corollary 13) Suppose is a closed hypersurface with and . Thenand

Here and is the region enclosed by . The equality occurs if and only if is a sphere centered at .

As another corollary, we have the following extension of Alexandrov’s theorem:

Corollary 3(Corollary 11) Suppose is a closed embedded hypersurface in . Assume , and there exists such that is constant where is the distance from . Then is a sphere.

The rest of this note is organized as follows. In Section 1 we will give the necessary definitions and preliminary results. In Section 2, we will prove the main results, and several corollaries will be given in Section 3.

**1. Preliminaries **

Let be an isometric immersion of an -dimensional semi-Riemannian manifold into an -dimensional semi-Riemannian manifold . We will use and to denote the connection on and respectively. The second fundamental form of in is defined by and is normal-valued. We denote by , where is a local orthonormal frame on . For simplicity, we will write as . For any normal vector field of in , we define the scalar second fundamental form by , and let .

We define the -th mean curvature as follows. If is even,

If is odd, the -th mean curvature is a normal vector field defined by

We also define . Here is zero if or for some , or if as sets, otherwise it is defined as the sign of the permutation . We also define the normalized -th mean curvature as

In the codimension one case, i.e. is a hypersurface, by taking the inner product with a unit normal if necessary, we can assume is scalar valued. In this case the value of is given by

where are the principal curvatures. This definition of will be used whenever is a hypersurface.

Following [Grosjean] and [Reilly1], we define the (generalized) -th Newton transformation of (as a tensor, possibly vector-valued) as follows.

If is even,

If is odd,

We also define . Again, in the codimension one case, by taking the inner product with a unit normal if necessary, we can assume is an ordinary tensor and if are the eigenvectors of , then

This definition of will be used whenever is a hypersurface.

We collect some basic properties of and :

Lemma 4We have ( denotes the trace on ):

*Proof:* These equations are well-known, at least in the codimension one case (e.g. [Mar] Lemma 2.1). They can be found e.g. in [Grosjean] Lemma 2.1, 2.2 and [K1] Lemma 2.1. For (4), if and , then by Codazzi equation, we have , which is equivalent to . The assertions are trivial for .

**2. Main results **

In this section, we will first derive a simple integral formula and draw a few direct consequences. We will use the notations in Section 1. Throughout this section, we will also assume that is a closed and oriented semi-Riemannian manifold isometrically immersed in , and denote the induced metric on by . We will omit the area element in the integrals when there is no confusion.

*Proof:* Let be the vector field defined by . Locally, let be a local orthonormal frame on such that , . Then , where and that . We can assume that for all . We compute the divergence of at :

The result follows by applying divergence theorem.

To proceed, let us recall that a vector field on is said to be a conformal (Killing) vector field if it satisfies

for some function on , and in this case, it is easy to see that . Here is the divergence on . More generally, for an immersion of into , a vector field is conformal along if for any -vector fields .

We now state and prove our first main result.

Theorem 6Suppose is a space form. Assume is a conformal vector field along with given by (2), and is a smooth function on .

*Proof:* Recall that . The result follows by applying Proposition 5 to , and using Lemma 4.

In general, it does not make sense to talk about if is odd. Even when the normal bundle is parallel so that makes sense, our approach does not seem to produce a result similar to that of Theorem 6, as we cannot produce the term and apply Lemma 4.

Instead, we will now take a different approach to derive a formula similar to (3) for odd , which is due to Strobing [S]. Similar to Section 1, for a family of normal vector fields (not necessarily distinct), we define

and

Similar to Lemma 4, we have

Lemma 7For , we have ( denotes the trace on ):

*Proof:* The proof is exactly the same as in the codimension one case of Lemma 4, see e.g. [Mar] Lemma 2.1, except that in (7), we need the fact that if is parallel.

By applying Proposition 5 to and using Lemma 7, we obtain the following

Theorem 8Suppose is a space form. Assume is a conformal vector field along with given by (2), is a smooth function on and are (not necessarily distinct) normal fields to which are parallel in the normal bundle. Then

Remark 1If , then Theorem 8 is reduced to (4) in Theorem 6.

**3. Some examples and applications **

By substituting different functions and in Theorem 6, we have several corollaries.

Corollary 9Suppose is immersed in . Then

- For all odd , we have Here we regard as a vector valued function.
- If is a hypersurface, then for all , we have Here we regard as a scalar.

*Proof:* This follows from Theorem 6 by putting and , , where are the standard orthonormal basis of . As are Killing vector fields, we have and the result follows. (Alternatively, this also follows from integrating the divergence of the vector field (more appropriately, an -tuple of vector fields) on , where is the position vector and regarded as an -tuple.)

Corollary 10Suppose is a closed hypersurface immersed in . Let be the position vector, , and is a smooth function on . Assume that , then we haveand

where .

*Proof:* The first equation follows from Theorem 6 and the observation that . The second equation follows by putting noting that .

We have the following extension of Alexandrov’s theorem.

*Proof:* Assume and is constant. Since has an elliptic point (i.e. point at which ), must be positive and hence . By [Mar] Proposition 3.2, then is positive definite and for . So by Corollary 10,

where is the region bounded by . Therefore

By Proposition 12 below, must be a sphere. If is injective, then is a positive constant as is constant. In particular, is star-shaped w.r.t. . Now, consider the furthest point and the nearest point on from , we have . We conclude that is centered at . The remaining cases can be proved similarly.

Proposition 12Suppose is a closed hypersurface embedded in such that , then

where is the region bounded by . The equality holds if and only if is a sphere.

*Proof:* By [Mar] Proposition 3.2, we have for all . In particular, by Newton’s inequalities, we have

By a result of Ros ([Ros] Theorem 1), we have The result follows.

If one of the equality holds then we conclude from the equality case of the Newton’s inequalities that is umbilical, and therefore is a sphere. If , then is also a sphere by [Ros] Theorem 1 again.

*Proof:* By [Mar] Proposition 3.2, if on , then is positive for . By applying Corollary 10 with and the Cauchy-Schwarz inequality, we can get the inequalities. If the equality holds, then as for , but then , which implies is a sphere centered at . The converse is easy. The inequality (6) follows from the fact that .

Remark 2Corollary 13 generalizes [K1] Theorem 3.2 (1) and also [K2] Theorem 2.

To state our next result, we first set up the notations. We define , , to be the vector space equipped with the semi-Riemannian metric . The position vector in will be denoted by and the inner product on by . Let and be a pseudo-sphere in . It is easy to see that is totally umbilic in and in particular has constant curvature.

Let us recall that the classical Hsiung-Minkowski formulas [Hsiung]: if is a space form and is a closed oriented hypersurface in with a unit normal vector field . Suppose possesses a conformal vector field , i.e. the Lie derivative of satisfies for some function , then we have

It is a nice observation that in general, if is a semi-Riemannian manifold which is isometrically embedded as a totally umbilic hypersuface in another semi-Riemannian manifold , and such that there exists a conformal vector field on , then the orthogonal projection of that vector field on is a conformal vector field on . Indeed, a simple calculation shows that on , if , then

Therefore is conformal on if is totally umbilic. In particular, we can construct a conformal vector field on by projecting any conformal vector field on onto .

In the following, we will consider the special case where the conformal vector field on is the orthogonal projection of a constant vector field on . More precisely, fix , considered as a parallel vector field on . The orthogonal projection of (this choice will make the conformal factor looks neater) on is then given by , or equivalently,

It is easily shown that the second fundamental form of in is

In particular, for defined in (8), in view of (7), we have

By Theorem 6, Theorem 8, and in view of (9), we have the following result:

Theorem 14Let be an -dimensional closed oriented semi-Riemannian manifold isometrically immersed in . Let be a smooth function on , be fixed and be given by (8).

- If is even, then
- If (i.e. hypersurface), then
Here and are scalars, is understood to be an ordinary -tensor, and is a unit normal vector field of in .

- If there exists (not necessarily distinct) normal fields to which are parallel in the normal bundle. Then

In the following, we will apply Theorem 14 to for different and . For simplicity, we will only give the result when is a hypersurface in (and consequently are scalars).

Let us consider the case where and so that . Choose and be the geodesic polar coordinates around on , where . Then

By Theorem 14 and the above, we have

Corollary 15With the notations above, let be a closed hypersurface in and be its unit normal. Suppose is a smooth function on . Then for ,

By substituting different functions in Corollary 15, we have the following corollary:

Corollary 16With the same assumptions as in Corollary 15, suppose and is contained in the open hemisphere centered at . Assume that , then we haveand

where .

*Proof:* The first equation follows by putting into Corollary 15 and observing that . The second equation follows by putting and noting that .

Corollary 17With the same assumptions as in Corollary 15, suppose and is contained in the open hemisphere centered at with .

The equality occurs if and only if is a sphere centered at .

*Proof:* By [Mar] Proposition 3.2, if on , then and are both positive for . Applying Corollary 15, we have

The equality (10) then follows by induction. If the equality case holds, then and so is a sphere centered at . The converse is easy.

For the case where and so that . We can choose and be the geodesic polar coordinates around on , where . Then

By Theorem 14, we have

Corollary 18With the notations above, let be a closed hypersurface in with unit normal vector . Suppose is a smooth function on , then for , we have

For the case where and so that , the de Sitter space. We choose and parametrize by , where . Then

By Theorem 14, we have

Corollary 19With the notations above, let () be a closed spacelike hypersurface in with unit normal vector . Suppose is a smooth function on , then for , we have

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