Not Every Horizontal Vector Field is Horizontal Lift

Theorem

Let $\pi : \R^2 \to \R$ be the projection map such that:

$\map \pi {x, y} = x$

Let $W = y \partial_x$ be a smooth vector field on $\R^2$.

Then $W$ is horizontal, but there is no smooth vector field whose horizontal lift is equal to $W$.

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics