Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...

, the Rauch comparison theorem, named after

Harry Rauch Harry Ernest Rauch (November 9, 1925 – June 18, 1979) was an American mathematician, who worked on complex analysis and differential geometry. He was born in Trenton, New Jersey, and died in White Plains, New York. Rauch earned his PhD i ...

, who proved it in 1951, is a fundamental result which relates the

sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...

of a

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 ...

to the rate at which

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

s spread apart. Intuitively, it states that for positive curvature, geodesics tend to converge, while for negative curvature, geodesics tend to spread. The statement of the theorem involves two Riemannian manifolds, and allows to compare the infinitesimal rate at which geodesics spread apart in the two manifolds, provided that their curvature can be compared. Most of the time, one of the two manifolds is a "comparison model", generally a manifold with

constant curvature In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature i ...

, and the second one is the manifold under study : a bound (either lower or upper) on its

is then needed in order to apply Rauch comparison theorem.

Statement

Let

M, \widetilde

be Riemannian manifolds, on which are drawn unit speed

segments

\gamma :

, T The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...

\to M and

\to \widetilde

. Assume that

\widetilde(0)

has no

conjugate points In differential geometry, conjugate points or focal points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian. Another viewpoi ...

along

\widetilde

, and let

J, \widetilde

be two normal

Jacobi field In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic for ...

s along

\gamma

and

\widetilde

such that : *

J(0) = 0

and

\widetilde(0) = 0

, D_t J(0),  = \left, \widetilde_t \widetilde(0)\

. If the sectional curvature of every 2-plane

\Pi \subset T_ M

containing

\dot(t)

is less or equal than the sectional curvature of every 2-plane

\widetilde \subset T_ \widetilde

containing

\dot(t)

, then

, J(t),  \geq , \widetilde(t),

for all

t \in

/math>.

Conditions of the theorem

The theorem is formulated using

s to measure the variation in geodesics. As the tangential part of a Jacobi field is independent of the geometry of the manifold, the theorem focuses on normal Jacobi fields, i.e. Jacobi fields which are orthogonal to the speed vector

\dot(t)

of the geodesic for all time

t

. Up to reparametrization, every variation of geodesics induces a normal Jacobi field. Jacobi fields are requested to vanish at time

t=0

because the theorem measures the infinitesimal divergence (or convergence) of a family of geodesics issued from the same point

\gamma(0)

, and such a family induces a Jacobi field vanishing at

t=0

Analog theorems

Under very similar conditions, it is also possible to compare the

Hessian A Hessian is an inhabitant of the German state of Hesse. Hessian may also refer to: Named from the toponym *Hessian (soldier), eighteenth-century German regiments in service with the British Empire **Hessian (boot), a style of boot **Hessian f ...

of the distance function to a given point. It is also possible to compare 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 this function (which is the trace of the Hessian), with some additional condition on one of the two manifolds: it is then enough to have an inequality on 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 ...

(which is the trace of the curvature tensor).

References

* do Carmo, M.P. ''Riemannian Geometry'', Birkhäuser, 1992. * Lee, J. M., ''Riemannian Manifolds: An Introduction to Curvature'', Springer, 1997. Theorems in Riemannian geometry {{differential-geometry-stub

Statement

Conditions of the theorem

Analog theorems

See also

References