Horizontal space

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Horizontal space is the term used in differential geometry for a particular choice of subspace of the set of all frames on the manifold. The horizontal space denoted by $H_b \in B$ (where $B$ is the space of all frames ) is one that obeys the following conditions.

$B b = H b + V b$
$H b g = g (H b)$
$H$ varies differentiably with $b$

where $V b$ is the vertical space. The collection of all such horizontal spaces ( $H : = {H b} b$ ) is called the connection.

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Categories: Differential geometry | Differential topology | Connection (mathematics)

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