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Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)
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Effect of the inverse exponential map on the curvature of a given curve
The curvature of $\alpha(t)$ is the same as the curvature of the curve $exp^{-1}_{\alpha(t)}(\alpha(s))$ at $s=t$. The previous curve is what you see from the tangent space of $\alpha(t)$. … How does the curvature change. I have done the computations for the curvature change in the case of a sphere and the result is a not very pleasant long formula. …