Definition:Vector Field along Smooth Curve
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Definition
Let $M$ be a smooth manifold with or without boundary.
Let $TM$ be the tangent bundle of $M$.
Let $I \subseteq \R$ be a real inverval.
Let $\gamma : I \to M$ be a smooth curve.
Let $V : I \to TM$ be a continuous map such that:
- $\forall t \in I : \map V t \in T_{\map \gamma t} M$
Then $V$ is called the vector field along (smooth) curve $\gamma$.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 4$: Connections. Vector and Tensor Fields Along Curves