Injectivity radius: Difference between revisions
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{{scalar function}} | {{scalar function}} | ||
{{radius notion}} | |||
''The term injectivity radius is also used for [[injectivity radius of a manifold]] which is the infimum over the manifold of the injectivity radii at all points'' | |||
==Definition== | ==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 | 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. | |||
Latest revision as of 19:47, 18 May 2008
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
The term injectivity radius is also used for injectivity radius of a manifold which is the infimum over the manifold of the injectivity radii at all points
Definition
The injectivity radius is a scalar function on a Riemannian manifold is defined as follows: the injectivity radius at is the supremum of all values such that the exponential map from the unit ball in , to the manifold , is injective.
The fact that the injectivity radius at each point is strictly positive is one of the starting points of Riemannian geometry.