Definition:Horizontal Vector Field
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Definition
Let $M$ be a smooth manifold.
Let $x \in M$ be a point.
Let $H_x$ be the horizontal tangent space.
Let $V$ be a vector field on $M$.
Suppose for each $x \in M$ the value of $V$ lies in the horizontal space of $x$:
- $\forall x \in M : \valueat V x \in H_x$
Then $V$ is called a horizontal vector field.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics