nLab
exponential map

Idea

The exponential function of classical analysis,

(1)$$\exp x≔\sum _{i=0}^{\mathrm{\infty }}\frac{x^i}{i!},$$\exp x \coloneqq \sum_{i = 0}^{\infty} \frac{x^i}{i!} ,

is the solution of the differential equation

$$f\prime =f$$f' = f

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:

• By dimensionalysis?, the argument of the function should be a tangent vector; so in the classical function $ℝ\to ℝ$, the source $ℝ$ is really the tangent space to the target $ℝ$ at the point $\exp 0=1$.
• We need a covariant derivative to tell us what $f\prime$ means.

So in the end we have, for any point $p$ on a differentiable manifold $M$ with an affine connection $\nabla$, a map ${\exp }_p:T_{p}M\to M$, 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 $f\prime =f$ 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 $\exp$:

$$b^x≔\exp (x\ln b).$$b^x \coloneqq \exp (x \ln b) .

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.

Definition

Let $M$ be a differentiable manifold, let $\nabla$ 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 maximally? defined partial function ${\exp }_p$ from the tangent space $T_{p}M$ to $M$ such that:

• $\nabla {\exp }_p={\exp }_p$ and
• ${\exp }_p(0)=1$.

This function is the natural exponential map on $M$ at $p$ relative to $\nabla$. We have ${\exp }_p:U\to M$, where $U$ is some neighbourhood of $0$ in $T_{p}M$. If $M$ is complete? (relative to $\nabla$), then $U$ will be all of $T_{p}M$.

Given any endomorphism $\varphi :T_{p}M\to T_{p}M$, we can also consider the exponential map on $M$ at $p$ relative to $\nabla$ with logarithmic base $\varphi$, which is simply $x\mapsto {\exp }_p\varphi (x)$. We say ‘logarithmic base’ since a classical exponential function with base $b$ corresponds to an exponential function whose logarithmic base is multiplication by $\ln b$.

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 naïvely, we may write

$$\gamma \prime =v,$$\gamma' = v ,

pretend that this is a differential equation for a function $\gamma :ℝ\to ℝ$, and take the solution

$$\gamma (t)=p\exp tx,$$\gamma(t) = p \exp t x ,

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

Let $M$ be a differentiable manifold, let $\nabla$ 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? and may be in any case), we have ${\exp }_{p}x=\gamma (1)$. In any case, we have ${\exp }_p(tx)=\gamma (t)$ for sufficiently small $t$.

In Riemannian manifolds

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 ${\nabla }_{\mathrm{lc}}$, so we define the natural Riemannian exponential map on $M$ at $p$ to be the natural exponential map on $M$ at $p$ relative to ${\nabla }_{\mathrm{lc}}$.

In Lie groups

Let $M$ be Lie group and let $𝔤$ 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 ${\nabla }_l$ or the canonical right-invariant connection ${\nabla }_r$. It turns out that the natural Riemannian exponential maps on $M$ at $1$ relative to ${\nabla }_l$ and ${\nabla }_r$ are the same; we define this to be the natural Lie exponential map on $M$ at the identity, denoted simply $\exp$. Several nice properties follow:

• $\exp$ is defined on all of $𝔤$.
• $\exp$ is a smooth homomorphism to $G$ from the underlying additive Lie group of $𝔤$.
• $\exp$ is surjective (a regular epimorphism) if $G$ is connected and compact (and also in some other situations, such as the classical cases where $G$ is $]0,\mathrm{\infty }[$ or $ℂ\setminus \{0\}$).
• If $G$ is compact, then it may be equipped with a Riemannian metric; then the Lie exponential map is the same as the Riemannian exponential map at $1$.
• If $G$ is a matrix Lie group, then $\exp$ is given by the classical series formula (1).

Logarithms

A logarithm is a partial section? of an exponential map.

Related concepts

• Hamiltonian flow