Definition:Main Curve

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

• Definition:Transverse Curve

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Geodesics and Minimizing Curves