# nLab vertical vector field

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Definition

Let $\pi : P \to X$ be a bundle in the category Diff of smooth manifolds. A vector field $v \in \Gamma(P)$ is vertical with respect to this bundle if it is in the kernel of the differential $d \pi : T P \to T X$.

A differential form on $P$ is a horizontal differential form with respect to $P \to X$ it it vanishes on vertical vector fields.

• parameterized index theorem?

Revised on September 18, 2013 22:14:07 by Urs Schreiber (145.116.129.172)