Suppose is an immersed orientable closed hypersurface. is the inner unit normal for and denote by the second fundamental form of the immersion and by , the principle curvatures at an arbitrary point of . The th mean curvature of is obtained by applying elementary symmetric function to . Equivalently, can be defined through the identity

for all real number . One can see that represents the mean curvature of , is the gauss-Kronecker curvature. can reflect the scalar curvature of on the condition that the ambient manifold is a space form.

We want to study the consequence of moving the hypersurface parallel. Namely, define to be

When is small enough, is well defined immersed hypersurface. Suppose are principle directions at a point of , then

here we identify as abbreviation. This implies that is also an unit normal field of . The area element will be

The second fundamental form of with respect to will be

for all tangent vector fields on . Plugging in and , we get

So are also principle directions for and principle curvatures are

Another way to see this is by choosing a geodesic local coordinates such that are the principle directions of at . Then

Since at . Therefore we get the principle curvature are .

Therefore the mean curvature for is

Since we have identity

which implies

Plugging in all the information,

Reorder the terms in the above identity by the order of , we get

One can use this to prove Heintze-Karcher inequality. There are Minkowski formula in Hyperbolic space and also.

**Remark:** S. Montiel and Anotnio Ros, compact hypersurfaces: the alexandrov theorem for higher order mean curvatures. Differential Geometry, 52, 279-296

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