In the mathematical field of

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, 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 submanifold In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...

s. It was discovered by

James Simons James Harris Simons (; born 25 April 1938) is an American mathematician, billionaire hedge fund manager, and philanthropist. He is the founder of Renaissance Technologies, a quantitative hedge fund based in East Setauket, New York. He and his f ...

in 1968. It can be viewed as a formula for the

Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...

of the second fundamental form of a

Riemannian submanifold A Riemannian submanifold ''N'' of a Riemannian manifold ''M'' is a submanifold of ''M'' equipped with the Riemannian metric inherited from ''M''. The image of an isometric immersion In mathematics, an embedding (or imbedding) is one instance of ...

. 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 of Euclidean space, the formula asserts that :

\Delta h=\operatornameH+Hh^2-, h, ^2h,

where, relative to a local choice of unit normal vector field, is the second fundamental form, is the mean curvature, and is the symmetric 2-tensor on given by . This has the consequence that :

\frac\Delta, h, ^2=, \nabla h, ^2-, h, ^4+\langle h,\operatornameH\rangle+H\operatorname(A^3)

where is the shape operator. In this setting, the derivation is particularly simple: :

\begin
\Delta h_&=\nabla^p\nabla_p h_\\
&=\nabla^p\nabla_ih_\\
&=\nabla_i\nabla^p h_-^qh_-^qh_\\
&=\nabla_i\nabla_jH-(h^h_-h_j^ph_i^q)h_-(h^h_-Hh_i^q)h_\\
&=\nabla_i\nabla_jH-, h, ^2h+Hh^2;
\end

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. In the even more general setting of arbitrary codimension, the formula involves a complicated polynomial in the second fundamental form.

References

Footnotes Books * * * Articles * * * *{{wikicite, ref={{sfnRef, Simons, 1968, reference=James Simons. ''Minimal varieties in Riemannian manifolds.'' Ann. of Math. (2) 88 (1968), 62–105. {{doi, 10.2307/1970556 {{closed access Differential geometry of surfaces Riemannian manifolds