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\section*{exponential map}

\begin{quote}%
For the concept in [[category theory]] see at \emph{[[exponential object]]}.


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{arithmetic}{}\paragraph*{{Arithmetic}}\label{arithmetic}

[[!include arithmetic geometry - contents]]

\hypertarget{exponential_maps}{}\section*{{Exponential maps}}\label{exponential_maps}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{with_respect_to_an_affine_connection}{With respect to an affine connection}\dotfill \pageref*{with_respect_to_an_affine_connection} \linebreak
\noindent\hyperlink{via_geodesics}{Via geodesics}\dotfill \pageref*{via_geodesics} \linebreak
\noindent\hyperlink{exp_of_Lie_groups}{In Lie groups}\dotfill \pageref*{exp_of_Lie_groups} \linebreak
\noindent\hyperlink{logarithms}{Logarithms}\dotfill \pageref*{logarithms} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The exponential [[function]] of classical [[analysis]] given by the [[series]],

\begin{equation}
\exp x \coloneqq \sum_{i = 0}^{\infty} \frac{x^i}{i!} ,
\label{series}\end{equation}
is the solution of the [[differential equation]]

\begin{displaymath}
f' = f
\end{displaymath}
with initial value $f(0) = 1$.

This classical function is defined on the [[real line]] (or the [[complex plane]]). To generalise it to other [[manifolds]], we need two things:

\begin{itemize}%
\item by its nature, the argument of the function should be a [[tangent vector]]; so in the classical function $\mathbb{R} \to \mathbb{R}$, the [[source]] $\mathbb{R}$ is really the [[tangent space]] to the [[target]] $\mathbb{R}$ at the point $\exp 0 = 1$.
\item We need a [[covariant derivative]] to tell us what $f'$ means.

\end{itemize}
So in the end we have, for any [[point]] $p$ on a [[differentiable manifold]] $M$ with an [[affine connection]] $\Del$, a map $\exp_p\colon T_p M \to M$, which is defined at least on a [[neighbourhood]] of $0$ in the [[tangent space]] $T_p M$.

Note that $p$ here comes from the initial value $\exp_p 0 = p$; we usually take $p = 1$ when we work in a [[Lie group]], but otherwise we are really generalising the classical exponential function $x \mapsto p \exp x$; every solution to $f' = f$ takes this form.

Classically, there are some other functions called `exponential'; given any nonzero real (or complex) number $b$, the map $x \mapsto b^x$ (or even $x \mapsto p\, b^x$) is also an exponential map. Using the [[natural logarithm]], we can define $b^x$ in terms of the natural exponential map $\exp$:

\begin{displaymath}
b^x \coloneqq \exp (x \ln b) .
\end{displaymath}
So while $b$ is traditionally called the `base', it is really the number $\ln b$ that matters, or even better the operation of multiplication by $\ln b$. This operation is an [[endomorphism]] of the real line (or complex plane), and every such endomorphism takes this form for some nonzero $b$ (and some branch of the natural logarithm, in the complex case). So we see that this generalised exponential map is simply the [[composite]] of the natural exponential map after a linear endomorphism.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

See also at \emph{[[flow of a vector field]]}.

\hypertarget{with_respect_to_an_affine_connection}{}\subsubsection*{{With respect to an affine connection}}\label{with_respect_to_an_affine_connection}

Let $M$ be a [[differentiable manifold]], let $\Del$ be an [[affine connection]] on $M$, and let $p$ be a [[point]] in $M$. Then by the general theory of [[differential equations]], there is a unique [[maximal partial function|maximally]] defined [[partial function]] $\exp_p$ from the [[tangent space]] $T_p M$ to $M$ such that:

\begin{itemize}%
\item $\Del \exp_p = \exp_p$ and
\item $\exp_p(0) = 1$.

\end{itemize}
This function is the \textbf{natural exponential map on $M$ at $p$ relative to $\Del$}. We have $\exp_p\colon U \to M$, where $U$ is some [[neighbourhood]] of $0$ in $T_p M$. If $M$ is [[complete manifold|complete]] (relative to $\Del$), then $U$ will be all of $T_p M$.

Let $M$ be a [[Riemannian manifold]] (or a [[pseudo-Riemannian manifold]]) and let $p$ be a point in $M$. Then $M$ may be equipped with the [[Levi-Civita connection]] $\Del_{lc}$, so we define the \textbf{natural Riemannian exponential map on $M$ at $p$} to be the natural exponential map on $M$ at $p$ relative to $\Del_{lc}$.

Given any [[endomorphism]] $\phi\colon T_p M \to T_p M$, we can also consider the \textbf{exponential map on $M$ at $p$ relative to $\Del$ with logarithmic base $\phi$}, which is simply $x \mapsto \exp_p \phi(x)$. We say `logarithmic base' since a classical exponential function with base $b$ corresponds to an exponential function whose logarithmic base is multiplication by $\ln b$.

\hypertarget{via_geodesics}{}\subsubsection*{{Via geodesics}}\label{via_geodesics}

Recall that a [[geodesic]] is a [[curve]] on a manifold whose [[velocity]] is constant (as measured along that curve relative to a given affine connection). Working na\"i{}vely, we may write

\begin{displaymath}
\gamma' = v ,
\end{displaymath}
pretend that this is a differential equation for a function $\gamma\colon \mathbb{R} \to \mathbb{R}$, and take the solution

\begin{displaymath}
\gamma(t) = p \exp t x ,
\end{displaymath}
where $p$ is given by the initial value $\gamma(0) = p$. We recognise this as being, morally, $\exp_p t x$. This suggests (although we need more work for a proof) the following result:

Let $M$ be a [[differentiable manifold]], let $\Del$ be an [[affine connection]] on $M$, and let $p$ be a [[point]] in $M$. Given a [[tangent vector]] $x$ at $p$, there is a unique [[maximal geodesic]] $\gamma$ on $M$ tangent to $x$ at $p$. If $\gamma(1)$ is defined (which it will be whenever $M$ is [[complete manifold|complete]] and may be in any case), we have $exp_p x = \gamma(1)$. In any case, we have $\exp_p (t x) = \gamma(t)$ for sufficiently small $t$.

\hypertarget{exp_of_Lie_groups}{}\subsection*{{In Lie groups}}\label{exp_of_Lie_groups}

Note: this section is under repair.

The classical exponential function $\exp \colon \mathbb{R} \to \mathbb{R}^*$ or $\exp \colon \mathbb{C} \to \mathbb{C}^*$ satisfies the fundamental property:

\begin{prop}
\label{}\hypertarget{}{}
The function $\exp \colon \mathbb{C} \to \mathbb{C}^*$ is a homomorphism taking addition to multiplication:

\begin{displaymath}
\exp(x + y) = \exp(x) \cdot \exp(y)
\end{displaymath}
\end{prop}
\begin{proof}
A number of proofs may be given. One rests on the combinatorial [[binomial theorem|binomial identity]]

\begin{displaymath}
(x + y)^n = \sum_{j + k = n} \frac{n!}{j! k!} x^j y^k
\end{displaymath}
(which crucially depends on the fact that multiplication is [[commutative monoid|commutative]]), whereupon

\begin{displaymath}
\itexarray{
\sum_{n \geq 0} \frac{(x+y)^n}{n!} & = & \sum_{n \geq 0} \sum_{j + k = n} \frac1{j!} \frac1{k!} x^j y^k \\
 & = & (\sum_{j \geq 0} \frac{x^j}{j!}) \cdot (\sum_{k \geq 0} \frac{y^k}{k!}) \\ 
 & = & \exp(x) \cdot \exp(y)
}
\end{displaymath}
An alternative proof begins with the observation that $f = \exp$ is the solution to the system $f' = f$, $f(0) = 1$. For each $y$, the function $g_1 \colon x \mapsto f(x) f(y)$ is a solution to the system $g' = g$, $g(0) = f(y)$, as is the function $g_2 \colon x \mapsto f(x + y)$. Then by uniqueness of solutions to ordinary differential equations (over connected domains; see, e.g., \href{http://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem}{here}), $g_1 = g_2$, i.e., $f(x + y) = f(x)f(y)$ for all $x, y$.\footnote{A previous edit offered even more detail: ``An alternative proof begins with the premise that each solution of the ordinary differential equation $g' = 0$ is locally constant. Suppose $c$ is a complex number. As $\exp' = \exp$, we find that $(\exp(x) \exp(c - x))' = \exp(x) \exp(c - x) + \exp(x) (-\exp(c-x)) = 0$. Hence, by the premise and the connectedness of the domain of $\exp$ (either ${\mathbb{R}}$ or ${\mathbb{C}}$), we obtain $\exp(x)\exp(c - x) = \exp(0)\exp(c)$. The initial condition $\exp(0) = 1$ then yields $\exp(x)\exp(c - x) = \exp(c)$. The result follows by setting $c = x + y$.''} 

\end{proof}
Let $M$ be [[Lie group]] and let $\mathfrak{g}$ be its [[Lie algebra]] $T_1 M$, the tangent space to the [[identity element]] $1$. Then $M$ may be equipped with the canonical left-invariant connection $\Del_l$ or the canonical right-invariant connection $\Del_r$. It turns out that the natural Riemannian exponential maps on $M$ at $1$ relative to $\Del_l$ and $\Del_r$ are the same; we define this to be the \textbf{natural Lie exponential map on $M$ at the identity}, denoted simply $\exp$. Several nice properties follow:

\begin{itemize}%
\item $\exp$ is defined on all of $\mathfrak{g}$.
\item $\exp \colon \mathfrak{g} \to G$ is a [[smooth map]].
\item If $\rho \colon \mathbb{R} \to \mathfrak{g}$ is a smooth homomorphism from the additive group $\mathbb{R}$ (i.e., if $\rho$ is an $\mathbb{R}$-linear map, uniquely determined by specifying $X = \rho(1)$), then $\exp \circ \rho_X \colon \mathbb{R} \to G$ is a smooth homomorphism.
\item For $X, Y \in \mathfrak{g}$, if $[X, Y] = 0$, then the restriction of $\exp \colon \mathfrak{g} \to G$ to the subspace spanned by $X$ and $Y$ is a smooth homomorphism to $G$. In particular, $\exp \colon \mathfrak{g} \to G$ is a homomorphism if $\mathfrak{g}$ is abelian (e.g., if $G$ is a commutative Lie group).
\item $\exp$ is [[surjection|surjective]] (a [[regular epimorphism]]) if $G$ is [[connected space|connected]] and [[compact space|compact]] (and also in some other situations, such as the classical cases where $G$ is $]0,\infty[$ or $\mathbb{C} \setminus \{0\}$). See \href{https://terrytao.wordpress.com/2011/06/25/two-small-facts-about-lie-groups/}{this post} by Terence Tao, Proposition 1; see also the first comment which indicates an alternative proof based on the fact that [[maximal tori]] in $G$ are all conjugate to one another. Note also that the exponential map might not be surjective if the compactness assumption is dropped, as in the case of $G = SL_2(\mathbb{R})$ or $SL_2(\mathbb{C})$, both of which are connected; see \href{http://math.stackexchange.com/questions/643216/non-surjectivity-of-the-exponential-map-to-sl2-mathbbc}{here} for instance.
\item If $G$ is [[compact space|compact]], then it may be equipped with a [[Riemannian metric]] that is both left and right invariant (see Tao's post linked in the previous remark); then the Lie exponential map is the same as the Riemannian exponential map at $1$.
\item If $G$ is a [[matrix Lie group]], then $\exp$ is given by the classical series formula \eqref{series}.

\end{itemize}
(to be expanded on)

\hypertarget{logarithms}{}\subsection*{{Logarithms}}\label{logarithms}

A \textbf{[[logarithm]]} is a [[local section]] of an exponential map.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[pane wave]]


\item [[flow of a vector field]]


\item [[Euler number]], [[e]]


\item [[logarithm]]


\item [[Euler's formula]]


\item [[exponential exact sequence]]


\item [[Hamiltonian flow]]


\item [[exponential modality]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item wikipedia \href{https://en.wikipedia.org/wiki/Exponential_map_%28Lie_theory%29}{exponential map (Lie theory)}, \href{https://en.wikipedia.org/wiki/Derivative_of_the_exponential_map}{derivative of the exponential map}, \href{https://en.wikipedia.org/wiki/Exponential_map_(Riemannian_geometry}{exponential map (Riemannian geometry)})
\item Springer [[eom]]: \href{https://www.encyclopediaofmath.org/index.php/Exponential_mapping}{exponential mapping}

\end{itemize}
An extensive treatment for the general exponential map for an affine connection, for exponential map for Riemannian manifolds and the one for Lie groups is

\begin{itemize}%
\item Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces

\end{itemize}
Specifically for Lie groups, a different detailed treatment of the exponential map is in

\begin{itemize}%
\item [[M M Postnikov]], \emph{Lie groups and Lie algebras}, Geometry V

\end{itemize}
Some nice historical notes are in

\begin{itemize}%
\item Wilfried Schmid, \emph{Poincare and Lie groups}, Bull. Amer. Math. Soc. 6:2, 1982 \href{http://www.ams.org/journals/bull/1982-06-02/S0273-0979-1982-14972-2/S0273-0979-1982-14972-2.pdf}{pdf}

\end{itemize}
[[!redirects exponential map]] [[!redirects exponential maps]] [[!redirects exponential function]] [[!redirects exponential functions]]

[[!redirects exponent]] [[!redirects exponents]]

[[!redirects exponentiation]] [[!redirects exponential]] [[!redirects exponentials]]

[[!redirects exponential series]]



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