Injectivity radius: Difference between revisions

Revision as of 11:53, 8 April 2007

This article defines a scalar function on a manifold, viz a function from the manifold to real numbers. The scalar function may be intrinsic or defined in terms of some other structure/functions

Definition

The injectivity radius is a scalar function on a Riemannian manifold $M$ is defined as follows: the injectivity radius at $x \in M$ is the supremum of all values $r$ such that the exponential map from the unit ball $B_{r} (x)$ in $T_{x} M$ , to the manifold $M$ , is injective.

The fact that the injectivity radius at each point is strictly positive is one of the starting points of Riemannian geometry.

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 ==Definition==
-The '''injectivity radius''' is a scalar function on a [[Riemannian manifold]] <math>M</math> is defined as follows: the injectivity radius at <math>x \in M</math> is the smallest value <math>r</math> such that the exponential map from the unit ball <math>B_r(x)</math> in <math>T_xM</math>, to the manifold <math>M</math>, is injective.
+The '''injectivity radius''' is a scalar function on a [[Riemannian manifold]] <math>M</math> is defined as follows: the injectivity radius at <math>x \in M</math> is the supremum of all values <math>r</math> such that the exponential map from the unit ball <math>B_r(x)</math> in <math>T_xM</math>, to the manifold <math>M</math>, is injective.
+The fact that the injectivity radius at each point is strictly positive is one of the starting points of Riemannian geometry.