Logarithm map

A local inverse to the exponential map near the identity of a Lie group.

Logarithm map

Let $G$ be a Lie group with Lie algebra $\mathfrak g$ .

Definition

A logarithm map on $G$ is a smooth map

\log:U\to \mathfrak g

defined on an open neighborhood $U$ of the identity element $e\in G$ such that

\exp(\log(g))=g \quad \text{for all } g\in U,

where $\exp:\mathfrak g\to G$ is the exponential map .

Equivalently, $\log$ is a (chosen) smooth local inverse to $\exp$ near $0\in\mathfrak g$ . Existence follows from the fact that $\exp$ is a local diffeomorphism at $0$ .

Basic properties

$\log(e)=0$ and $d(\log)_e$ is the inverse of $d(\exp)_0=\mathrm{id}_{\mathfrak g}$ .
$\log$ is generally not globally defined or single-valued on all of $G$ (even for compact groups).

Concrete calculation (matrix groups)

If $G\subset \mathrm{GL}(n,\Bbb R)$ is a matrix Lie group and $A$ is sufficiently close to $I$ , one has the convergent power series

\log(I+X)=\sum_{k=1}^\infty \frac{(-1)^{k+1}}{k}X^k \quad\text{for }\|X\|\text{ small},

so for $A$ near $I$ ,

\log(A)=\log\bigl(I+(A-I)\bigr)=\sum_{k=1}^\infty \frac{(-1)^{k+1}}{k}(A-I)^k.

In particular, for $X\in \mathfrak g$ and $t$ small, $\log(\exp(tX))=tX$ .

Context

The logarithm is useful for “linearizing” group multiplication near the identity. For example, the product of two near-identity elements can be expressed in $\mathfrak g$ using the Baker–Campbell–Hausdorff formula , which compares $\log(gh)$ to $\log(g)+\log(h)$ with Lie bracket corrections.