In this post I will show a matrix inequality by Huang and Wu, and give one geometric application of it. See their paper for other applications.
Proposition 1 (Huang-Wu) For an matrix, ,
where , and .
In particular if is real and ,
Proof: Denote by . Consider
We want to eliminate in the above. This is done by observing (summation runs from if not specified)
Substitute this expression of into the first equation, we are done.
Let be a Riemannian manifold and be a -dimensional submanifold of . Let be a local orthnormal frame on such that are tangential to . Denote by and the scalar curvature and the Ricci curvature of respectively, and to be the mean curvature and the shape operator of in respectively.
Corollary 3 With the notations and assumptions above, if on , then
The equality holds if and only if at that point, and if .
Proof: Clearly the inequality holds at any point where . Suppose , then by Proposition 2 we have at . But the Gauss equation shows that . Hence the inequality holds.
The necessary and sufficient condition for the equality can be easily seen from the RHS of the equation in Proposition 2.
When is a hypersurface in , then the condition in Corollary 3 can be replaced by the extrinsic curvature conditions of . Let be the principal curvatures of in . We define the -th mean curvature of to be
For even , is an intrinsic quantity (by Reilly) depending only on the induced metric of . For odd , is well-defined up to a sign (i.e. the choice of the normal ), but is independent of the choice of .
Proposition 4 With the same notations as before, if is nonnegative and
then on . In particular, in this case, .
Proof: To simplify notations, we simply denote by . There is an orthonormal which diagonalizes the shape operator: . Then is diagonalized by (using Gauss equation) with
Let . Then by Newton’s identities, we have
Plugging these into (2), and using , we have
The result follows.
Remark 1 The condition (1) in Proposition 4 is automatically satisfied when ( is defined to be zero if ). However the inequality is trivial in this case, as is a curve, and is just the geodesic curvature of . In general, (1) can also be replaced by other stronger condition. For example, by a Maclaurin-type inequality (see Hardy, Littlewood, Polya’s Inequalities p.52 Theorem 53, or here for a slightly stronger statement)
so we can replace the condition by .