Given a Riemannian manifold \((M,g)\) and

\begin{equation*} \mathcal{E}=\{v\in TM: \gamma_v \text{ is defined on an interval containing \([0,1]\).}\}, \end{equation*}

where \(\gamma_v\) is defined as discussed in (0x66dee47d) . The exponential map \(\exp\colon \mathcal{E}\to M\) is defined by

\begin{equation*} \exp(v)=\gamma_v(1). \end{equation*}

The exponential map \(\exp_p\) for a point \(p\in M\) is defined as the restriction on \(\mathcal{E}_p=\mathcal{E}\cap T_pM\).

Properties Link to heading

  • The exponential map is smooth.
  • The differential \(d(\exp_p)_0\) is the identity on \(T_pM\).
  • The exponential map is locally diffeomorph because of the remark above.

See also Link to heading