The injectivity domain at a point Riemannian
manifold is the set of tangent vectors
that reach a point before the cut time along their geodesic .
In terms of the exponential map , the open injectivity
domain is
Its closure includes minimizing vectors at the cut time. The domain is star-shaped about zero, but need not
be convex for an arbitrary metric.
See also Exponential Map ,
Geodesic ,
Ma-Trudinger-Wang Condition ,
Riemannian
Metric ,
Tangent Space
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References Figalli, A.; Gallouët, T. O.; and Rifford, L. "On the Convexity of Injectivity Domains on Nonfocal Manifolds." SIAM
J. Math. Anal. 47 , 969-1000, 2015. https://doi.org/10.1137/140961821 .
Cite this as:
Weisstein, Eric W. "Injectivity Domain."
From MathWorld https://mathworld.wolfram.com/InjectivityDomain.html
Subject classifications
of a complete Riemannian
manifold is the set of tangent vectors
that reach a point before the cut time along their geodesic.
In terms of the exponential map, the open injectivity
domain is
