In the mathematical field of

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, 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 that ...

s. It was discovered by James Simons 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 th ...

of the

second fundamental form In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundamen ...

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 of

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

, the formula asserts that :

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

where, relative to a local choice of unit normal vector field, is the

, is the

mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The ...

, 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 mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspective ...

. 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 Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mos ...

. 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