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In mathematics, Bochner's formula is a statement relating
harmonic functions In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, ...
on a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent spac ...
(M, g) to the
Ricci curvature In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
. The formula is named after the American
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
Salomon Bochner Salomon Bochner (20 August 1899 – 2 May 1982) was an Austrian mathematician, known for work in mathematical analysis, probability theory and differential geometry. Life He was born into a Jewish family in Podgórze (near Kraków), then Au ...
.


Formal statement

If u \colon M \rightarrow \mathbb is a smooth function, then : \tfrac12 \Delta, \nabla u, ^2 = g(\nabla\Delta u,\nabla u) + , \nabla^2 u, ^2 + \mbox(\nabla u, \nabla u) , where \nabla u is the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of u with respect to g, \nabla^2 u is the Hessian of u with respect to g and \mbox is the Ricci curvature tensor.. If u is harmonic (i.e., \Delta u = 0 , where \Delta=\Delta_g is 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 ...
with respect to the metric g ), Bochner's formula becomes : \tfrac12 \Delta, \nabla u, ^2 = , \nabla^2 u, ^2 + \mbox(\nabla u, \nabla u) . Bochner used this formula to prove the Bochner vanishing theorem. As a corollary, if (M, g) is a Riemannian manifold without boundary and u \colon M \rightarrow \mathbb is a smooth, compactly supported function, then : \int_M (\Delta u)^2 \, d\mbox = \int_M \Big( , \nabla^2 u, ^2 + \mbox(\nabla u, \nabla u) \Big) \, d\mbox . This immediately follows from the first identity, observing that the integral of the left-hand side vanishes (by the divergence theorem) and integrating by parts the first term on the right-hand side.


Variations and generalizations

* Bochner identity * Weitzenböck identity


References

{{reflist Differential geometry