3

Depending on the curvature of a certain manifold $M$, intuitively it is interesting to quantify how quickly normal coordinates defined at a point $p$ deteriorate away from $p$. Is there a standardized measure to quantify the actual size of the neighbourhood of $p$ the normal coordinates are valid in? If so, how is it calculated?

Kagaratsch
  • 2,279
  • 3
    The standard term is the injectivity radius at $p$. This essentially tells you the largest ball under which normal coordinates are still well defined. – J.V.Gaiter Feb 09 '22 at 22:42
  • 1
    @J.V.Gaiter Thank you for the clarification! For the benefit of other visitors, following this terminology I found the following file explaining it: https://homepage.univie.ac.at/james.grant/papers/NullInj/Inj_rad_talk_1.pdf – Kagaratsch Feb 09 '22 at 22:49
  • 1
    There is also the conjugate radius, which can be estimated below using a lower bound of the sectional curvature. – Deane Feb 10 '22 at 01:43

0 Answers0