Definition:Tangential Acceleration of Smooth Curve

Definition

Let $I \subseteq \R$ be a real interval.

Let $M \subseteq \R^n$ be an embedded submanifold.

Let $\gamma: \R \to M$ be a smooth curve.

Let $\map {\gamma} t$ be the acceleration of $\gamma$ at $t \in I$.

Let $T_{\map \gamma t} M$ be the tangent space at $\map \gamma t \in M$ for some $t \in I$.

Let $\pi^\top : T_{\map \gamma t} \R^n \to T_{\map \gamma t} M$ be the tangential projection.

The tangential acceleration of $\gamma$ at $t \in I$ is defined as:

$\map {\gamma} t^\top := \map {\pi^\top} {\map {\gamma} t}$

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 4$: Connections. The Problem of Differentiating Vector Fields