In this short note, we will prove the following generalization of Hsiung-Minkowski formulas for hypersurfaces in the Euclidean space.

Theorem 1Let be a closed hypersurface, then for any and , we have

Here is the normalized -th mean curvature, , is the tangential component of the position vector onto . (If , we use the convention that and the second term on the RHS of this equation is understood to be zero. )

The classical Hsiung-Minkowski formulas [Hsiung] can be recovered by putting in the above equation. This also generalizes the results in one of my previous posts.

** 0.1. Preliminaries **

Let us fix some notations. Let be a hypersurface and . Let be a local orthonormal frame on and let be the unit outward normal of . Let and be the connections on and respectively. We define the the shape operator by and is defined by . By abusing of notation, we will also denote the second fundamental form by .

We define the -th mean curvature and the normalized -th mean curvature of by

respectively. We use the convention that .

Following [Reilly], we define the -th Newton transformation of as

Alternatively, can be defined recursively by (see e.g. [Reilly])

We use the convention that , for example. Unless otherwise stated, repeated indices will be summed over .

We will need the following two lemmas.

Lemma 2For all , we have the following identities:

*Proof:* The equations (2) and (3) require the Codazzi equation which holds for any hypersurface in a space form. (2) is well-known and can be found e.g. in [Reilly]. To prove (1), let us assume that are the principal directions with principal curvatures .

We now prove (3). Note that by (1) and (2), we have

The third line follows from the fact that is symmetric in the indices , .

Lemma 3For , we have

*Proof:* See e.g. [LWX] Lemma 2.

** 0.2. Main result **

We can now state and prove our main result.

Theorem 4Let be a closed hypersurface, then for any and , we have

Here is the tangential component of the position vector onto .

*Proof:* We first assume . In the following computation, we will omit in the integral symbol. Using Lemma 2 and integration by parts, we compute

where we have used Lemma 3. Using , we have

Here we have used (1) in the last line. Substitute this into (2), we obtain

This is equivalent to

We now consider the case where . Define the function on . Then it is easy to check that where is the Euclidean metric. Let be a local orthonormal frame on , and be the restriction of on . Then we have and . From this we see that

So we have

Therefore

This completes the proof.

Corollary 5For , suppose is convex (i.e. ), then we haveIn particular,

and

Here is the region bounded by .