Simons' formula

Mathematical formula From Wikipedia, the free encyclopedia

In the mathematical field of differential geometry, the Simons formula (also known as the Simons identity, and in some variants as the Simons inequality) is a fundamental equation in the study of minimal submanifolds. It was discovered by James Simons in 1968.[1] It can be viewed as a formula for the Laplacian of the second fundamental form of a Riemannian submanifold. It is often quoted and used in the less precise form of a formula or inequality for the Laplacian of the length of the second fundamental form.

In the case of a hypersurface M of Euclidean space, the formula asserts that

where, relative to a local choice of unit normal vector field, h is the second fundamental form, H is the mean curvature, and h2 is the symmetric 2-tensor on M given by h2
ij
= gpqhiphqj
.[2] This has the consequence that

where A is the shape operator.[3] In this setting, the derivation is particularly simple:

the only tools involved are the Codazzi equation (equalities #2 and 4), the Gauss equation (equality #4), and the commutation identity for covariant differentiation (equality #3). The more general case of a hypersurface in a Riemannian manifold requires additional terms to do with the Riemann curvature tensor.[4] In the even more general setting of arbitrary codimension, the formula involves a complicated polynomial in the second fundamental form.[5]

References

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