Definition:Main Curve
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Definition
Let $\struct {M, g}$ be a Riemannian manifold.
Let $I, J \subseteq \R$ be real intervals.
Let $\Gamma : J \times I \to M$ be a one-parameter family of curves, where $\times$ denotes the cartesian product.
Let $s \in J$ be constant.
Then for all $t \in I$ the map $\map {\Gamma_s} t = \map \Gamma {s, t}$ is called the main curve.
Also see
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Geodesics and Minimizing Curves