Definition:Tangential Directional Derivative

Definition

Let $S \subseteq \R^n$ be an open subset of $\R^n$.

Let $M \subseteq S$ be an embedded submanifold.

Let $p \in S$ be a point.

Let $T_p S$ be the tangent space at $p \in S$.

Let $v \in T_p S$ be a vector.

Let $\map {\mathfrak{X}} M$ and $\map {\mathfrak{\tilde X}} S$ be the spaces of smooth vector fields of $M$ and $S$ respectively.

Let $Y \in \map {\mathfrak{X}} {M}$ and $\tilde Y \in \map {\mathfrak{X}} S$ be a vector fields.

Let $\pi^\top : T_p S \to T_p M$ be the tangential projection.

Let $\nabla_v {\tilde Y}$ be Euclidean directional derivative of $\tilde Y$ in the direction $v$.

The tangential directional derivative of $Y$ in the direction $v$ is defined by:

$\nabla_v^\top Y := \map {\pi^\top} {\bar \nabla_v \tilde Y}$

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