mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

— specifically,

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 Bochner identity is an identity concerning

harmonic map In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a ...

s between

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

s. The identity is named after the American mathematician Salomon Bochner.

Statement of the result

Let ''M'' and ''N'' be

s and let ''u'' : ''M'' → ''N'' be a harmonic map. Let d''u'' denote the derivative (pushforward) of ''u'', ∇ the gradient, Δ the

Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named af ...

, Riem_''N'' the Riemann curvature tensor on ''N'' and Ric_''M'' the Ricci curvature tensor on ''M''. Then :

\frac12 \Delta \big( ,  \nabla u , ^ \big) = \big,  \nabla ( \mathrm u ) \big, ^ + \big\langle \mathrm_ \nabla u, \nabla u \big\rangle - \big\langle \mathrm_ (u) (\nabla u, \nabla u) \nabla u, \nabla u \big\rangle.

References

External links

* Differential geometry Mathematical identities {{differential-geometry-stub

Statement of the result

See also

References

External links