# Exponential mapping

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A mapping of the tangent space of a manifold $M$ into $M$. It is defined by a connection given on $M$ and is a far-reaching generalization of the ordinary exponential function regarded as a mapping of a straight line into itself.
1) Let $M$ be a $C ^ \infty$- manifold with an affine connection, let $p$ be a point in $M$, let $M _ {p}$ be the tangent space to $M$ at $p$, let $X$ be a non-zero vector in $M _ {p}$, and let $t \rightarrow \gamma _ {X} ( t)$ be the geodesic passing through $p$ in the direction of $X$. There is an open neighbourhood $N _ {0}$ of the point $0$ in $M _ {p}$ and an open neighbourhood $N _ {p}$ of $p$ in $M$ such that the mapping $X \rightarrow \gamma _ {X} ( 1)$ is a diffeomorphism of $N _ {0}$ onto $N _ {p}$. This mapping is called the exponential mapping at $p$ and is denoted by $\mathop{\rm exp}$. A neighbourhood $N _ {0}$ is called normal if: 1) the mapping $\mathop{\rm exp}$ maps $N _ {0}$ onto $N _ {p}$ diffeomorphically; and 2) $X \in N _ {0}$ and $0 \leq t \leq 1$ imply that $t X \in N _ {0}$. In this case $N _ {p}$ is said to be a normal neighbourhood of the point $p$ in the manifold $M$. Every $p \in M$ has a convex normal neighbourhood $N _ {p}$: Any two points of such a neighbourhood can be joined by exactly one geodesic segment lying in $N _ {p}$. If $M$ is a complete Riemannian manifold, then $\mathop{\rm exp}$ is a surjective mapping of $M _ {p}$ onto $M$.
2) Let $G$ be a Lie group with identity $e$ and let $\mathfrak g$ be the corresponding Lie algebra consisting of the tangent vectors to $G$ at $e$. For every vector $X \in \mathfrak g$ there is a unique differentiable homomorphism $\theta$ of the group $\mathbf R$ into $G$ such that the tangent vector to $\theta ( \mathbf R )$ at $e$ coincides with $X$. The mapping $X \rightarrow \mathop{\rm exp} X = \theta ( 1)$ is called the exponential mapping of the algebra $\mathfrak g$ into the group $G$. There is an open neighbourhood $N _ {0}$ of the point $0$ in $\mathfrak g$ and an open neighbourhood $N _ {e}$ of $e$ in $G$ such that $\mathop{\rm exp}$ is a diffeomorphism of $N _ {0}$ onto $N _ {e}$. Let $X _ {1} \dots X _ {n}$ be some basis for the algebra $\mathfrak g$. The mapping $\mathop{\rm exp} ( x _ {1} X _ {1} + {} \dots + x _ {n} X _ {n} ) \rightarrow ( x _ {1} \dots x _ {n} )$ is a coordinate system on $N _ {e}$; these coordinates are called canonical.
The concept of an exponential mapping of a Lie group $G$ can also be approached from another point of view. There is a one-to-one correspondence between the set of all affine connections on $G$ that are invariant relative to the group of left translations and the set of bilinear functions $\alpha : \mathfrak g \times \mathfrak g \rightarrow \mathfrak g$. It turns out that the exponential mapping $\mathop{\rm exp}$ of the algebra $\mathfrak g$ into the group $G$ coincides with the mapping $\mathop{\rm exp}$ of the tangent space of $\mathfrak g$ into the manifold $G$ at the point $e$ in this manifold with respect to the left-invariant affine connection corresponding to any skew-symmetric bilinear function $\alpha$.