Abstract
In this paper, we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold [Formula: see text]: (1) the convexity radius of [Formula: see text], [Formula: see text], where [Formula: see text] is the injectivity radius of [Formula: see text] and [Formula: see text] is the focal radius of open ball centered at [Formula: see text] with radius [Formula: see text]; (2) for any two points [Formula: see text] in [Formula: see text], [Formula: see text] where [Formula: see text] is the conjugate radius of [Formula: see text]; (3) for any [Formula: see text], any (not necessarily minimizing) geodesic in [Formula: see text] has length [Formula: see text]. We also clarify two different concepts on convexity radius and give examples to illustrate that the one more frequently used in literature is not continuous.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.