The Exponential Map for Lie Groups

Godement, Roger

doi:10.1007/978-3-319-54375-8_6

The Exponential Map for Lie Groups

Roger Godement⁹

Chapter
First Online: 11 May 2017

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Part of the book series: Universitext ((UTX))

Abstract

The exponential map for Lie groups is studied and used to prove properties of Lie groups and the relation with Lie algebras.

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Notes

1.
Let R be a vector field on a manifold G; assume that both R and G are of class ${C}^r$ with $r=\infty $ or $\omega $. The integral of (6.1.8) is then defined to be any pair $(I, \gamma ) $ where I is an open interval in ${\mathbb R}$ and $\gamma : I \rightarrow G$ is of class ${C}^r$ and satisfies (6.1.8) for all $ t \in I$. Replacing G and R by the domain of a local chart and the restriction of R to it, locally, the search for integrals of (6.1.8) amounts to integrating a system of ordinary differential equations
$$dx_i/dt = R_i(x_1,\ldots , x_n)$$
in an open subset of a Cartesian space; it follows immediately that (a) every ${C}^1$ solution of (6.1.8) is in fact ${C}^r$, (b) if two integrals $(I, \gamma ) $ and $(J, \delta )$ of (6.1.8) are equal at some point of $ I \cap J$, they are equal in the neighbourhood of this point and so in all of $I\cap J$ since $I\cap J $ is connected, (c) every integral can be extended to a maximal integral, i.e. which cannot be extended to a strictly larger interval, (d) for all $ a \in G$ and $ s\in \mathbb R$, there is a unique maximal integral of (6.1.8) satisfying $\gamma (s) = a$. See M. Berger and B. Gostiaux, Differential Geometry, Chap. I, or H. Cartan, Calcul Différentiel, Chap. II (where, for the needs of our presentation, the Banach spaces can be assumed to be finite-dimensional), or J. Dieudonné, Eléments d’Analyse, X.4, X.5, X.7 and XVIII.1 and XVIII.2, or MA IX, Sect. 15, etc.
2.
We let the reader recover formula (3.9.14) of Chap. 3 from this result.
3.
The Campbell–Hausdorff series of $\mathfrak {g}$ and $\mathfrak {h}$ are written $H_{\mathfrak {g}}$ and $H_{\mathfrak {h}}$.
4.
Bull. of the Physico-Mathematics Society of Kazan, 1935. Ado’s complicated original proof has since then been greatly simplified.
5.
If we were willing to use Ado’s theorem, which we are not, then we would observe that by taking the composite of j and a local embedding of G into a linear group gives a local embedding of H into a linear group; Theorem 9 would then follow from the analogous statement in Chap. 3.
6.
Exercise: extend Theorem 7 to the homomorphisms of Chap. 2, Sect. 2.2.
7.
We give some brief indications about this point of view, leaving aside all superfluous subtleties. Let G be a ${C}^r$ manifold with $r=\infty $ or $r=\omega $, and suppose that at each point x of G, there is vector subspace $\mathfrak {h}(x)$ of some fixed dimension p of the tangent vector space to G at x. Suppose that the map $ x\mapsto \mathfrak {h}(x)$ is of class ${C}^r$ in the following sense: in every sufficiently small open subset U of G, there exist p vector fields of class ${C}^r$ on U whose values, at each point x of U, form a basis of the corresponding subspace $\mathfrak {h}(x)$.
An integral manifold of the field $\mathfrak {h}$ of subspaces is defined to be any submanifold H of G such that
$$\begin{aligned}T_x(H) = \mathfrak {h}(x)\quad \text { for all } \quad x \in H,\end{aligned}$$
and $\mathfrak {h}$ is said to be integrable if there is an integral submanifold passing through every point of G. Frobenius’ Theorem affirms that this is the case if and only if, for all open subsets $U\subset G$ and all vector fields L and M on U,
$$\begin{aligned}L(x), M(x) \in \mathfrak {h}(x)\quad \text { for all }\quad x \in U\quad \text { implies }\end{aligned}$$

$$\begin{aligned}{}[L, M](x) \in \mathfrak {h}( x)\quad \text { for all }\quad x \in U. \qquad {(*)}\end{aligned}$$
Besides it suffices to check that when there are vector fields $L_i$ $ (1\le i\le p)$ on an open subset U, which, at each $x\in U$, form a basis of $\mathfrak {h}(x)$, there exist functions $c^k_{ij}$ on U such that
$$\begin{aligned}{}[L_i, L_j] =\sum c^k_{ij} L_k\qquad {(*)}\end{aligned}$$
for all i and j.
When $\mathfrak {h}$ is integrable, G can be endowed with a new manifold structure ($G(\mathfrak {h})$ will denote the manifold obtained by endowing the set G with this new structure of class ${C}^r$ as in the case of G and $\mathfrak {h}$) having the following properties: the identity map from $G(\mathfrak {h})$ to G is an immersion; its tangent map at each point $x\in G$ maps the tangent space to $G(\mathfrak {h})$ at x isomorphically onto the given subspace $\mathfrak {h} (x)$ of $ T_x(G)$. The connected components of the manifold $G(\mathfrak {h})$ are called the maximal integral submanifolds of $\mathfrak {h}$; they are of dimension p and each of them is canonically endowed with an injective immersion into G, but in general they cannot be identified with any veritable submanifold of G. To apply these results to the theory of Lie groups, take G to be a Lie group and $\mathfrak {h}$ the field of subspaces $x\mapsto x\cdot \mathfrak {h}$, where $\mathfrak {h}$ is a fixed Lie subalgebra of the Lie algebra $\mathfrak {g}$ of G. The integrability condition trivially holds and the maximal integral submanifold through the point e is then just the “Lie”, “analytic”, “integral” or “immersed” subgroup of G defined by $\mathfrak {h}$, endowed with the Lie group structure of Theorem 10.
Most presentations of differential geometry prove Frobenius’ Theorem. For example, see M. Berger and B. Gostiaux, Differential Geometry, (Springer, 1988) or Cl. Godbillon, Géométrie différentielle et mécanique analytique (Herman, 1969).
8.
See Theorem 15 below.
9.
In the statement of Theorem 4 of Chap. 4, the continuity assumption on h is in fact superfluous. Indeed, if the map $h\circ \pi $ is a morphism, then it is perforce continuous: but a submersion is an open map; the continuity of $h\circ \pi $ on X then implies the continuity of h on the open subset $\pi (X) $ of Y.
10.
As H is a normal subgroup of K, $gHk = gkH$ for all $ g \in G $ and $k \in K$. This is why K can be made to act on the right on G / H. The actions thus obtained on the group G / H are obviously right translations by elements of the subgroup p(K).

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Roger Godement

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Godement, R. (2017). The Exponential Map for Lie Groups. In: Introduction to the Theory of Lie Groups. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-54375-8_6

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DOI: https://doi.org/10.1007/978-3-319-54375-8_6
Published: 11 May 2017
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-54373-4
Online ISBN: 978-3-319-54375-8
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