Category:Exponential Maps
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This category contains results about Exponential Maps.
Definitions specific to this category can be found in Definitions/Exponential Maps.
Let $\struct {M, g, \nabla}$ be a Riemannian or pseudo-Riemannian manifold without boundary endowed with the Levi-Civita connection.
Let $T_p M$ be the tangent space of $M$ at $p \in M$.
Let $v \in T_p M$.
Let $I \subseteq \R$ be an open real interval.
Let $\gamma_v : I \to M$ be the unique maximal geodesic such that:
- $\map {\gamma '} 0 = v$
where $\gamma'$ is the velocity of $\gamma$.
Let $TM$ be the tangent bundle of $M$.
Let $\EE \subseteq TM$ be the set defined as:
- $\EE = \set {v \in TM : \text{$\gamma_v$ is defined on $I : \closedint 0 1 \subseteq I$}}$
Then the exponential map, denoted by $\exp$, is the mapping $\exp : \EE \to M$ such that:
- $\map \exp v = \map {\gamma_v} 1$
Pages in category "Exponential Maps"
The following 2 pages are in this category, out of 2 total.