Exponential map (Lie theory)

<h2 id="definitions">Definitions</h2>
Let 
 
 
 
 G
 
 
 {\displaystyle G}
 
 be a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> and 
 
 
 
 
 
 g
 
 
 
 
 {\displaystyle {\mathfrak {g}}}
 
 be its <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> (thought of as the <a href="/facts/Tangent_space/crvpbSeG">tangent space</a> to the <a href="/facts/Identity_element/9nss3xHY">identity element</a> of 
 
 
 
 G
 
 
 {\displaystyle G}
 
). The exponential map is a map

exp
        :
        
          
            g
          
        
        →
        G
      
    
    {\displaystyle \exp \colon {\mathfrak {g}}\to G}

which can be defined in several different ways. The typical modern definition is this:

Definition: The exponential of 
 
 
 
 X
 ∈
 
 
 g
 
 
 
 
 {\displaystyle X\in {\mathfrak {g}}}
 
 is given by 
 
 
 
 exp
 ⁡
 (
 X
 )
 =
 γ
 (
 1
 )
 
 
 {\displaystyle \exp(X)=\gamma (1)}
 
 where

γ
        :
        
          R
        
        →
        G
      
    
    {\displaystyle \gamma \colon \mathbb {R} \to G}

is the unique <a href="/facts/One-parameter_subgroup/Ep7zASA0">one-parameter subgroup</a> of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 whose <a href="/facts/Tangent_vector/UEq5yffm">tangent vector</a> at the identity is equal to 
 
 
 
 X
 
 
 {\displaystyle X}
 
.
It follows easily from the <a href="/facts/Chain_rule/aMLYxP0x">chain rule</a> that 
 
 
 
 exp
 ⁡
 (
 t
 X
 )
 =
 γ
 (
 t
 )
 
 
 {\displaystyle \exp(tX)=\gamma (t)}
 
. The map 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
, a group homomorphism from 
 
 
 
 (
 
 R
 
 ,
 +
 )
 
 
 {\displaystyle (\mathbb {R} ,+)}
 
 to 
 
 
 
 G
 
 
 {\displaystyle G}
 
, may be constructed as the <a href="/facts/Integral_curve/wGbVcymt">integral curve</a> of either the right- or left-invariant <a href="/facts/Vector_field/ccFNuPPp">vector field</a> associated with 
 
 
 
 X
 
 
 {\displaystyle X}
 
. That the integral curve exists for all real parameters follows by right- or left-translating the solution near zero.
We have a more concrete definition in the case of a <a href="/facts/Matrix_Lie_group/a9jCQyjF">matrix Lie group</a>. The exponential map coincides with the <a href="/facts/Matrix_exponential/WID5cG2g">matrix exponential</a> and is given by the ordinary series expansion:

exp
 ⁡
 (
 X
 )
 =
 
 ∑
 
 k
 =
 0
 
 
 ∞
 
 
 
 
 
 X
 
 k
 
 
 
 k
 !
 
 
 
 =
 I
 +
 X
 +
 
 
 1
 2
 
 
 
 X
 
 2
 
 
 +
 
 
 1
 6
 
 
 
 X
 
 3
 
 
 +
 ⋯
 
 
 {\displaystyle \exp(X)=\sum _{k=0}^{\infty }{\frac {X^{k}}{k!}}=I+X+{\frac {1}{2}}X^{2}+{\frac {1}{6}}X^{3}+\cdots }
 
,
where 
 
 
 
 I
 
 
 {\displaystyle I}
 
 is the <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>. Thus, in the setting of matrix Lie groups, the exponential map is the restriction of the matrix exponential to the Lie algebra 
 
 
 
 
 
 g
 
 
 
 
 {\displaystyle {\mathfrak {g}}}
 
 of 
 
 
 
 G
 
 
 {\displaystyle G}
 
.

<h3>Comparison with Riemannian exponential map</h3>
If 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is compact, it has a Riemannian metric invariant under left and right translations, then the Lie-theoretic exponential map for 
 
 
 
 G
 
 
 {\displaystyle G}
 
 coincides with the <a href="/facts/Exponential_map_(Riemannian_geometry)/YfsKa30B">exponential map of this Riemannian metric</a>.
For a general 
 
 
 
 G
 
 
 {\displaystyle G}
 
, there will not exist a Riemannian metric invariant under both left and right translations. Although there is always a Riemannian metric invariant under, say, left translations, the exponential map in the sense of Riemannian geometry for a left-invariant metric will not in general agree with the exponential map in the Lie group sense. That is to say, if 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is a Lie group equipped with a left- but not right-invariant metric, the geodesics through the identity will not be one-parameter subgroups of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 .

<h3>Other definitions</h3>
Other equivalent definitions of the Lie-group exponential are as follows:

<ul><li>It is the exponential map of a canonical left-invariant <a href="/facts/Affine_connection/WRjjxSlM">affine connection</a> on G, such that <a href="/facts/Parallel_transport/QX0Wfcte">parallel transport</a> is given by left translation. That is, 
 
 
 
 exp
 ⁡
 (
 X
 )
 =
 γ
 (
 1
 )
 
 
 {\displaystyle \exp(X)=\gamma (1)}
 
 where 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
 is the unique <a href="/facts/Geodesic/fbWR6l80">geodesic</a> with the initial point at the identity element and the initial velocity X (thought of as a tangent vector).</li>
<li>It is the exponential map of a canonical right-invariant affine connection on G. This is usually different from the canonical left-invariant connection, but both connections have the same geodesics (orbits of 1-parameter subgroups acting by left or right multiplication) so give the same exponential map.</li>
<li>The <a href="/facts/Lie_group%25E2%2580%2593Lie_algebra_correspondence/zAnzPAV1">Lie group–Lie algebra correspondence</a> also gives the definition: for 
 
 
 
 X
 ∈
 
 
 g
 
 
 
 
 {\displaystyle X\in {\mathfrak {g}}}
 
, the mapping 
 
 
 
 t
 ↦
 exp
 ⁡
 (
 t
 X
 )
 
 
 {\displaystyle t\mapsto \exp(tX)}
 
 is the unique Lie group homomorphism 
 
 
 
 (
 
 R
 
 ,
 +
 )
 →
 G
 
 
 {\displaystyle (\mathbb {R} ,+)\to G}
 
 corresponding to the Lie algebra homomorphism 
 
 
 
 
 R
 
 →
 
 
 g
 
 
 
 
 {\displaystyle \mathbb {R} \to {\mathfrak {g}}}
 
, 
 
 
 
 t
 ↦
 t
 X
 .
 
 
 {\displaystyle t\mapsto tX.}
 
</li>
<li>The exponential map is characterized by the differential equation 
 
 
 
 
 
 d
 
 d
 t
 
 
 
 exp
 ⁡
 (
 t
 X
 )
 =
 exp
 ⁡
 (
 t
 X
 )
 ⋅
 X
 
 
 {\textstyle {\frac {d}{dt}}\exp(tX)=\exp(tX)\cdot X}
 
, where the right side uses the translation mapping 
 
 
 
 
 
 g
 
 
 =
 
 T
 
 e
 
 
 G
 →
 
 T
 
 g
 
 
 G
 ,
  
 X
 ↦
 g
 ⋅
 X
 
 
 {\displaystyle {\mathfrak {g}}=T_{e}G\to T_{g}G,\ X\mapsto g\cdot X}
 
 for 
 
 
 
 g
 =
 exp
 ⁡
 (
 t
 X
 )
 
 
 {\displaystyle g=\exp(tX)}
 
. In the one-dimensional case, this is equivalent to 
 
 
 
 
 exp
 ′
 
 ⁡
 (
 x
 )
 =
 exp
 ⁡
 (
 x
 )
 
 
 {\displaystyle \exp '(x)=\exp(x)}
 
.</li></ul>
<h2 id="examples">Examples</h2>
<ul><li>The <a href="/facts/Unit_circle/VsTffjPK">unit circle</a> centered at 0 in the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> is a Lie group (called the <a href="/facts/Circle_group/kuzdWtHo">circle group</a>) whose tangent space at 1 can be identified with the imaginary line in the complex plane, 
 
 
 
 {
 i
 t
 :
 t
 ∈
 
 R
 
 }
 .
 
 
 {\displaystyle \{it:t\in \mathbb {R} \}.}
 
 The exponential map for this Lie group is given by</li></ul>

i
        t
        ↦
        exp
        ⁡
        (
        i
        t
        )
        =
        
          e
          
            i
            t
          
        
        =
        cos
        ⁡
        (
        t
        )
        +
        i
        sin
        ⁡
        (
        t
        )
        ,
        
      
    
    {\displaystyle it\mapsto \exp(it)=e^{it}=\cos(t)+i\sin(t),\,}

that is, the same formula as the ordinary <a href="/facts/Complex_exponential/Wq7VXIV0">complex exponential</a>.
<ul><li>More generally, for <a href="/facts/Complex_torus/nh9BNgYU">complex torus</a><a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>: 8  
 
 
 
 X
 =
 
 
 C
 
 
 n
 
 
 
 /
 
 Λ
 
 
 {\displaystyle X=\mathbb {C} ^{n}/\Lambda }
 
 for some integral <a href="/facts/Lattice_(group)/2ls9Iv9N">lattice</a> 
 
 
 
 Λ
 
 
 {\displaystyle \Lambda }
 
 of rank 
 
 
 
 n
 
 
 {\displaystyle n}
 
 (so isomorphic to 
 
 
 
 
 
 Z
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {Z} ^{n}}
 
) the torus comes equipped with a <a href="/facts/Universal_Cover/bDRGasl7">universal covering map</a></li></ul>
<blockquote>
 
 
 
 π
 :
 
 
 C
 
 
 n
 
 
 →
 X
 
 
 {\displaystyle \pi :\mathbb {C} ^{n}\to X}
 
</blockquote>from the quotient by the lattice. Since 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is locally isomorphic to 
 
 
 
 
 
 C
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {C} ^{n}}
 
 as <a href="/facts/Complex_manifold/DlvPnpzm">complex manifolds</a>, we can identify it with the tangent space 
 
 
 
 
 T
 
 0
 
 
 X
 
 
 {\displaystyle T_{0}X}
 
, and the map<blockquote>
 
 
 
 π
 :
 
 T
 
 0
 
 
 X
 →
 X
 
 
 {\displaystyle \pi :T_{0}X\to X}
 
</blockquote>corresponds to the exponential map for the complex Lie group 
 
 
 
 X
 
 
 {\displaystyle X}
 
.
<ul><li>In the <a href="/facts/Quaternions/T3tnu7mX">quaternions</a> 
 
 
 
 
 H
 
 
 
 {\displaystyle \mathbb {H} }
 
, the set of <a href="/facts/Versor/TTalepkk">quaternions of unit length</a> form a Lie group (isomorphic to the special unitary group SU(2)) whose tangent space at 1 can be identified with the space of purely imaginary quaternions, 
 
 
 
 {
 i
 t
 +
 j
 u
 +
 k
 v
 :
 t
 ,
 u
 ,
 v
 ∈
 
 R
 
 }
 .
 
 
 {\displaystyle \{it+ju+kv:t,u,v\in \mathbb {R} \}.}
 
 The exponential map for this Lie group is given by</li></ul>

w
        
        :=
        (
        i
        t
        +
        j
        u
        +
        k
        v
        )
        ↦
        exp
        ⁡
        (
        i
        t
        +
        j
        u
        +
        k
        v
        )
        =
        cos
        ⁡
        (
        
          |
        
        
          w
        
        
          |
        
        )
        1
        +
        sin
        ⁡
        (
        
          |
        
        
          w
        
        
          |
        
        )
        
          
            
              w
            
            
              
                |
              
              
                w
              
              
                |
              
            
          
        
        .
        
      
    
    {\displaystyle \mathbf {w} :=(it+ju+kv)\mapsto \exp(it+ju+kv)=\cos(|\mathbf {w} |)1+\sin(|\mathbf {w} |){\frac {\mathbf {w} }{|\mathbf {w} |}}.\,}

This map takes the 2-sphere of radius R inside the purely imaginary <a href="/facts/Quaternion/T3tnu7mX">quaternions</a> to 
 
 
 
 {
 s
 ∈
 
 S
 
 3
 
 
 ⊂
 
 H
 
 :
 Re
 ⁡
 (
 s
 )
 =
 cos
 ⁡
 (
 R
 )
 }
 
 
 {\displaystyle \{s\in S^{3}\subset \mathbf {H} :\operatorname {Re} (s)=\cos(R)\}}
 
, a 2-sphere of radius 
 
 
 
 sin
 ⁡
 (
 R
 )
 
 
 {\displaystyle \sin(R)}
 
 (cf. <a href="/facts/Pauli_matrices/BdeC2wos">Exponential of a Pauli vector</a>). Compare this to the first example above.
<ul><li>Let V be a finite dimensional real vector space and view it as a Lie group under the operation of vector addition. Then 
 
 
 
 Lie
 ⁡
 (
 V
 )
 =
 V
 
 
 {\displaystyle \operatorname {Lie} (V)=V}
 
 via the identification of V with its tangent space at 0, and the exponential map</li></ul>

exp
        :
        Lie
        ⁡
        (
        V
        )
        =
        V
        →
        V
      
    
    {\displaystyle \operatorname {exp} :\operatorname {Lie} (V)=V\to V}

is the identity map, that is, 
 
 
 
 exp
 ⁡
 (
 v
 )
 =
 v
 
 
 {\displaystyle \exp(v)=v}
 
.
<ul><li>In the <a href="/facts/Split-complex_number/DtavqChA">split-complex number</a> plane 
 
 
 
 z
 =
 x
 +
 y
 ȷ
 ,
 
 
 ȷ
 
 2
 
 
 =
 +
 1
 ,
 
 
 {\displaystyle z=x+y\jmath ,\quad \jmath ^{2}=+1,}
 
 the imaginary line 
 
 
 
 {
 ȷ
 t
 :
 t
 ∈
 
 R
 
 }
 
 
 {\displaystyle \lbrace \jmath t:t\in \mathbb {R} \rbrace }
 
 forms the Lie algebra of the <a href="/facts/Unit_hyperbola/AvwG6gJ9">unit hyperbola</a> group 
 
 
 
 {
 cosh
 ⁡
 t
 +
 ȷ
  
 sinh
 ⁡
 t
 :
 t
 ∈
 
 R
 
 }
 
 
 {\displaystyle \lbrace \cosh t+\jmath \ \sinh t:t\in \mathbb {R} \rbrace }
 
 since the exponential map is given by</li></ul>

ȷ
        t
        ↦
        exp
        ⁡
        (
        ȷ
        t
        )
        =
        cosh
        ⁡
        t
        +
        ȷ
         
        sinh
        ⁡
        t
        .
      
    
    {\displaystyle \jmath t\mapsto \exp(\jmath t)=\cosh t+\jmath \ \sinh t.}

<h2 id="properties">Properties</h2>
<h3>Elementary properties of the exponential</h3>
For all 
 
 
 
 X
 ∈
 
 
 g
 
 
 
 
 {\displaystyle X\in {\mathfrak {g}}}
 
, the map 
 
 
 
 γ
 (
 t
 )
 =
 exp
 ⁡
 (
 t
 X
 )
 
 
 {\displaystyle \gamma (t)=\exp(tX)}
 
 is the unique <a href="/facts/One-parameter_subgroup/Ep7zASA0">one-parameter subgroup</a> of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 whose <a href="/facts/Tangent_vector/UEq5yffm">tangent vector</a> at the identity is 
 
 
 
 X
 
 
 {\displaystyle X}
 
. It follows that:

<ul><li>
 
 
 
 exp
 ⁡
 (
 (
 t
 +
 s
 )
 X
 )
 =
 exp
 ⁡
 (
 t
 X
 )
 exp
 ⁡
 (
 s
 X
 )
 
 
 
 {\displaystyle \exp((t+s)X)=\exp(tX)\exp(sX)\,}
 
</li>
<li>
 
 
 
 exp
 ⁡
 (
 −
 X
 )
 =
 exp
 ⁡
 (
 X
 
 )
 
 −
 1
 
 
 .
 
 
 
 {\displaystyle \exp(-X)=\exp(X)^{-1}.\,}
 
</li></ul>
More generally:

<ul><li>
 
 
 
 exp
 ⁡
 (
 X
 +
 Y
 )
 =
 exp
 ⁡
 (
 X
 )
 exp
 ⁡
 (
 Y
 )
 ,
 
 
 if 
 
 [
 X
 ,
 Y
 ]
 =
 0
 
 
 {\displaystyle \exp(X+Y)=\exp(X)\exp(Y),\quad {\text{if }}[X,Y]=0}
 
.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li></ul>
The preceding identity does not hold in general; the assumption that 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 commute is important.
The image of the exponential map always lies in the <a href="/facts/Identity_component/rFCx5kro">identity component</a> of 
 
 
 
 G
 
 
 {\displaystyle G}
 
.

<h3>The exponential near the identity</h3>
The exponential map 
 
 
 
 exp
 :
 
 
 g
 
 
 →
 G
 
 
 {\displaystyle \exp \colon {\mathfrak {g}}\to G}
 
 is a <a href="/facts/Smooth_map/mffL4ch2">smooth map</a>. Its <a href="/facts/Pushforward_(differential)/xLsoLwVH">differential</a> at zero, 
 
 
 
 
 exp
 
 ∗
 
 
 :
 
 
 g
 
 
 →
 
 
 g
 
 
 
 
 {\displaystyle \exp _{*}\colon {\mathfrak {g}}\to {\mathfrak {g}}}
 
, is the identity map (with the usual identifications).
It follows from the inverse function theorem that the exponential map, therefore, restricts to a <a href="/facts/Diffeomorphism/855N5YYj">diffeomorphism</a> from some neighborhood of 0 in 
 
 
 
 
 
 g
 
 
 
 
 {\displaystyle {\mathfrak {g}}}
 
 to a neighborhood of 1 in 
 
 
 
 G
 
 
 {\displaystyle G}
 
.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
It is then not difficult to show that if G is connected, every element g of G is a product of exponentials of elements of 
 
 
 
 
 
 g
 
 
 
 
 {\displaystyle {\mathfrak {g}}}
 
:<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
 
 
 
 g
 =
 exp
 ⁡
 (
 
 X
 
 1
 
 
 )
 exp
 ⁡
 (
 
 X
 
 2
 
 
 )
 ⋯
 exp
 ⁡
 (
 
 X
 
 n
 
 
 )
 ,
 
 
 X
 
 j
 
 
 ∈
 
 
 g
 
 
 
 
 {\displaystyle g=\exp(X_{1})\exp(X_{2})\cdots \exp(X_{n}),\quad X_{j}\in {\mathfrak {g}}}
 
.
Globally, the exponential map is not necessarily surjective. Furthermore, the exponential map may not be a local diffeomorphism at all points. For example, the exponential map from 
 
 
 
 
 
 s
 o
 
 
 
 
 {\displaystyle {\mathfrak {so}}}
 
(3) to <a href="/facts/Rotation_group_SO(3)/yXbKPzhT">SO(3)</a> is not a local diffeomorphism; see also <a href="/facts/Cut_locus_(Riemannian_manifold)/DBsbljDw">cut locus</a> on this failure. See <a href="/facts/Derivative_of_the_exponential_map/1gkglfwg">derivative of the exponential map</a> for more information.

<h3>Surjectivity of the exponential</h3>
In these important special cases, the exponential map is known to always be surjective:

<ul><li>G is connected and compact,<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li>
<li>G is connected and nilpotent (for example, G connected and abelian), or</li>
<li>
 
 
 
 G
 =
 G
 
 L
 
 n
 
 
 (
 
 C
 
 )
 
 
 {\displaystyle G=GL_{n}(\mathbb {C} )}
 
.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a></li></ul>
For groups not satisfying any of the above conditions, the exponential map may or may not be surjective.
The image of the exponential map of the connected but non-compact group <a href="/facts/SL2(R)/xjKM9JLJ">SL2(R)</a> is not the whole group. Its image consists of C-diagonalizable matrices with eigenvalues either positive or with modulus 1, and of non-diagonalizable matrices with a repeated eigenvalue 1, and the matrix 
 
 
 
 −
 I
 
 
 {\displaystyle -I}
 
. (Thus, the image excludes matrices with real, negative eigenvalues, other than 
 
 
 
 −
 I
 
 
 {\displaystyle -I}
 
.)<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h3>Exponential map and homomorphisms</h3>
Let 
 
 
 
 ϕ
 :
 G
 →
 H
 
 
 {\displaystyle \phi \colon G\to H}
 
 be a Lie group homomorphism and let 
 
 
 
 
 ϕ
 
 ∗
 
 
 
 
 {\displaystyle \phi _{*}}
 
 be its <a href="/facts/Pushforward_(differential)/xLsoLwVH">derivative</a> at the identity. Then the following diagram <a href="/facts/Commutative_diagram/bdG8xuer">commutes</a>:<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

In particular, when applied to the <a href="/facts/Adjoint_representation_of_a_Lie_group/bWG4c2su">adjoint action</a> of a Lie group 
 
 
 
 G
 
 
 {\displaystyle G}
 
, since 
 
 
 
 
 Ad
 
 ∗
 
 
 =
 ad
 
 
 {\displaystyle \operatorname {Ad} _{*}=\operatorname {ad} }
 
, we have the useful identity:<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

A
 d
 
 
 exp
 ⁡
 X
 
 
 (
 Y
 )
 =
 exp
 ⁡
 (
 
 
 a
 d
 
 
 X
 
 
 )
 (
 Y
 )
 =
 Y
 +
 [
 X
 ,
 Y
 ]
 +
 
 
 1
 
 2
 !
 
 
 
 [
 X
 ,
 [
 X
 ,
 Y
 ]
 ]
 +
 
 
 1
 
 3
 !
 
 
 
 [
 X
 ,
 [
 X
 ,
 [
 X
 ,
 Y
 ]
 ]
 ]
 +
 ⋯
 
 
 {\displaystyle \mathrm {Ad} _{\exp X}(Y)=\exp(\mathrm {ad} _{X})(Y)=Y+[X,Y]+{\frac {1}{2!}}[X,[X,Y]]+{\frac {1}{3!}}[X,[X,[X,Y]]]+\cdots }
 
.
<h2 id="logarithmic-coordinates">Logarithmic coordinates</h2>
Given a Lie group 
 
 
 
 G
 
 
 {\displaystyle G}
 
 with Lie algebra 
 
 
 
 
 
 g
 
 
 
 
 {\displaystyle {\mathfrak {g}}}
 
, each choice of a basis 
 
 
 
 
 X
 
 1
 
 
 ,
 …
 ,
 
 X
 
 n
 
 
 
 
 {\displaystyle X_{1},\dots ,X_{n}}
 
 of 
 
 
 
 
 
 g
 
 
 
 
 {\displaystyle {\mathfrak {g}}}
 
 determines a coordinate system near the identity element e for G, as follows. By the <a href="/facts/Inverse_function_theorem/pIwcyc6X">inverse function theorem</a>, the exponential map 
 
 
 
 exp
 :
 N
 
 
 →
 ∼
 
 
 U
 
 
 {\displaystyle \operatorname {exp} :N{\overset {\sim }{\to }}U}
 
 is a diffeomorphism from some neighborhood 
 
 
 
 N
 ⊂
 
 
 g
 
 
 ≃
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle N\subset {\mathfrak {g}}\simeq \mathbb {R} ^{n}}
 
 of the origin to a neighborhood 
 
 
 
 U
 
 
 {\displaystyle U}
 
 of 
 
 
 
 e
 ∈
 G
 
 
 {\displaystyle e\in G}
 
. Its inverse:

log
        :
        U
        
          
            →
            ∼
          
        
        N
        ⊂
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \log :U{\overset {\sim }{\to }}N\subset \mathbb {R} ^{n}}

is then a coordinate system on U. It is called by various names such as logarithmic coordinates, exponential coordinates or normal coordinates. See the <a href="/facts/Closed-subgroup_theorem/uxS593Uy">closed-subgroup theorem</a> for an example of how they are used in applications.
Remark: The open cover 
 
 
 
 {
 U
 g
 
 |
 
 g
 ∈
 G
 }
 
 
 {\displaystyle \{Ug|g\in G\}}
 
 gives a structure of a <a href="/facts/Real-analytic_manifold/EN1fKI1r">real-analytic manifold</a> to G such that the group operation 
 
 
 
 (
 g
 ,
 h
 )
 ↦
 g
 
 h
 
 −
 1
 
 
 
 
 {\displaystyle (g,h)\mapsto gh^{-1}}
 
 is real-analytic.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/List_of_exponential_topics/TNgs72qw">List of exponential topics</a></li>
<li><a href="/facts/Derivative_of_the_exponential_map/1gkglfwg">Derivative of the exponential map</a></li>
<li><a href="/facts/Matrix_exponential/WID5cG2g">Matrix exponential</a></li></ul>
<h2 id="citations">Citations</h2>

<h2 id="works-cited">Works cited</h2>

<ul><li>Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3319134666.</li>
<li>Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, <a href="/facts/Graduate_Studies_in_Mathematics/APstozeU">Graduate Studies in Mathematics</a>, vol. 34, Providence, R.I.: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-2848-9, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1834454">1834454</a>.</li>
<li>Kobayashi, Shoshichi; Nomizu, Katsumi (1996), <a href="/facts/Foundations_of_Differential_Geometry/lgGBjhhd">Foundations of Differential Geometry</a>, vol. 1 (New ed.), Wiley-Interscience, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-15733-3.</li>
<li><a href="https://www.encyclopediaofmath.org/index.php?title=Exponential_mapping">"Exponential mapping"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Birkenhake, Christina (2004). Complex Abelian Varieties. Herbert Lange (Second, augmented ed.). Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-662-06307-1. OCLC 851380558. <a href="978-3-662-06307-1" target="_blank">978-3-662-06307-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">This follows from the Baker-Campbell-Hausdorff formula. <a href="/wiki/Baker-Campbell-Hausdorff_formula" target="_blank">/wiki/Baker-Campbell-Hausdorff_formula</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Hall 2015 Corollary 3.44 - Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Hall 2015 Corollary 3.47 - Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Hall 2015 Corollary 11.10 - Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Hall 2015 Exercises 2.9 and 2.10 - Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Hall 2015 Exercise 3.22 - Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Hall 2015 Theorem 3.28 - Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Hall 2015 Proposition 3.35 - Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Kobayashi & Nomizu 1996, p. 43. - Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, vol. 1 (New ed.), Wiley-Interscience, ISBN 0-471-15733-3 <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
</ol>

Exponential map (Lie theory) open-in-new

Exponential map (Lie theory)