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.
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