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 ...

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 Aust ...

proved in 1946 that any

Killing vector field In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal gene ...

of a compact

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 ...

with negative

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 measure ...

must be zero. Consequently the

isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...

of the manifold must be finite.

Discussion

The theorem is a corollary of Bochner's more fundamental result which says that on any connected Riemannian manifold of negative Ricci curvature, the length of a nonzero

cannot have a local maximum. In particular, on a closed Riemannian manifold of negative Ricci curvature, every Killing vector field is identically zero. Since the isometry group of a complete Riemannian manifold is a

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

whose

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

is naturally identified with the vector space of Killing vector fields, it follows that the isometry group is zero-dimensional. Bochner's theorem then follows from the fact that the isometry group of a closed Riemannian manifold is compact. Bochner's result on Killing vector fields is an application of the

maximum principle In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. ...

as follows. As an application of the Ricci commutation identities, the formula :

\Delta X=-\nabla(\operatornameX)+\operatorname(\mathcal_Xg)-\operatorname(X,\cdot)

holds for any vector field on a pseudo-Riemannian manifold.In an alternative notation, this says that

\nabla^p\nabla_pX_i=-\nabla_i\nabla^pX_p+\nabla^p(\nabla_iX_p+\nabla_pX_i)-R_X^p.

As a consequence, there is :

\frac\Delta\langle X,X\rangle=\langle\nabla X,\nabla X\rangle-\nabla_X\operatornameX+\langle X,\operatorname(\mathcal_Xg)\rangle-\operatorname(X,X).

In the case that is a Killing vector field, this simplifies to :

\frac\Delta\langle X,X\rangle=\langle\nabla X,\nabla X\rangle-\operatorname(X,X).

In the case of a Riemannian metric, the left-hand side is nonpositive at any local maximum of the length of . However, on a Riemannian metric of negative Ricci curvature, the right-hand side is strictly positive wherever is nonzero. So if has a local maximum, then it must be identically zero in a neighborhood. Since Killing vector fields on connected manifolds are uniquely determined from their value and derivative at a single point, it follows that must be identically zero.

Notes

References

* * * * * * * * Theorems in differential geometry {{differential-geometry-stub