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| {{about|the exponential map in differential geometry|discrete dynamical systems|Exponential map (discrete dynamical systems)}}
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| [[File:Azimuthal Equidistant N90.jpg|thumb|right|The exponential map of the Earth as viewed from the north pole is the polar [[azimuthal equidistant projection]] in cartography.]] | |
| In [[differential geometry]], the '''exponential map''' is a generalization of the ordinary [[exponential function]] of mathematical analysis to all differentiable manifolds with an [[affine connection]]. Two important special cases of this are the exponential map for a manifold with a [[Riemannian metric]], and the exponential map from a [[Lie algebra]] to a [[Lie group]].
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| == Definition ==
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| Let ''M'' be a [[differentiable manifold]] and ''p'' a point of ''M''. An [[affine connection]] on ''M'' allows one to define the notion of a [[geodesic]] through the point ''p''.<ref>A source for this section is {{harvtxt|Kobayashi|Nomizu|1975|loc=§III.6}}, which uses the term "linear connection" where we use "affine connection" instead.</ref>
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| Let ''v'' ∈ T<sub>''p''</sub>''M'' be a [[tangent vector]] to the manifold at ''p''. Then there is a unique [[geodesic]] γ<sub>''v''</sub> satisfying γ<sub>''v''</sub>(0) = ''p'' with initial tangent vector γ′<sub>''v''</sub>(0) = ''v''. The corresponding '''exponential map''' is defined by exp<sub>''p''</sub>(''v'') = γ<sub>v</sub>(1). In general, the exponential map is only ''locally defined'', that is, it only takes a small neighborhood of the origin at T<sub>''p''</sub>''M'', to a neighborhood of ''p'' in the manifold. This is because it relies on the theorem on [[Picard–Lindelöf theorem|existence and uniqueness]] for [[ordinary differential equation]]s which is local in nature. An affine connection is called complete if the exponential map is well-defined at every point of the [[tangent bundle]].
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| == Lie theory ==
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| {{Lie groups |Algebras}}
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| In the theory of [[Lie group]]s, the '''exponential map''' is a map from the [[Lie algebra]] of a Lie group to the group which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary justifications for the study of Lie groups at the level of Lie algebras.
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| The ordinary [[exponential function]] of mathematical analysis is a special case of the exponential map when ''G'' is the multiplicative group of non-zero [[real number]]s (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects.
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| ===Definitions===
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| Let <math>G</math> be a [[Lie group]] and <math>\mathfrak g</math> be its [[Lie algebra]] (thought of as the [[tangent space]] to the [[identity element]] of <math>G</math>). The '''exponential map''' is a map
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| :<math>\exp\colon \mathfrak g \to G</math> | |
| which can be defined in several different ways as follows:
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| *It is the exponential map of a canonical left-invariant affine connection on ''G'', such that parallel transport is given by left translation.
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| *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.
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| *It is given by <math>\exp(X) = \gamma(1)</math> where
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| :<math>\gamma\colon \mathbb R \to G</math>
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| :is the unique [[one-parameter subgroup]] of <math>G</math> whose [[tangent vector]] at the identity is equal to <math>X</math>. It follows easily from the [[chain rule]] that <math>\exp(tX) = \gamma(t)</math>. The map <math>\gamma</math> may be constructed as the [[integral curve]] of either the right- or left-invariant [[vector field]] associated with <math>X</math>. That the integral curve exists for all real parameters follows by right- or left-translating the solution near zero.
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| * If <math>G</math> is a [[matrix Lie group]], then the exponential map coincides with the [[matrix exponential]] and is given by the ordinary series expansion:
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| ::<math>\exp (X) = \sum_{k=0}^\infty\frac{X^k}{k!} = I + X + \frac{1}{2}X^2 + \frac{1}{6}X^3 + \cdots</math>
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| :(here <math>I</math> is the [[identity matrix]]).
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| *If ''G'' is compact, it has a Riemannian metric invariant under left and right translations, and the exponential map is the exponential map of this Riemannian metric.
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| ===Examples===
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| * The [[unit circle]] centered at 0 in the [[complex plane]] is a Lie group (called the [[circle group]]) whose tangent space at 1 can be identified with the imaginary line in the complex plane, <math>\{it:t\in\mathbb R\}.</math> The exponential map for this Lie group is given by
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| *: <math>it \mapsto \exp(it) = e^{it} = \cos(t) + i\sin(t),\,</math>
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| : that is, the same formula as the ordinary [[complex exponential]].
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| * In the [[split-complex number]] plane <math>z = x + y \jmath , \quad \jmath^2 = +1,</math> the imaginary line <math>\lbrace \jmath t : t \in \mathbb R \rbrace</math> forms the Lie algebra of the [[unit hyperbola]] group <math>\lbrace \cosh t + \jmath \ \sinh t : t \in \mathbb R \rbrace</math> since the exponential map is given by
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| *:<math>\jmath t \mapsto \exp(\jmath t) = \cosh t + \jmath \ \sinh t.</math>
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| * The unit 3-sphere <math>S^3</math> centered at 0 in the [[quaternions]] '''H''' is a Lie group (isomorphic to the special unitary group <math>SU(2)</math>) whose tangent space at 1 can be identified with the space of purely imaginary quaternions, <math>\{it+ju + kv :t, u, v\in\mathbb R\}.</math> The exponential map for this Lie group is given by
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| *: <math>\bold{w} = (it+ju+kv) \mapsto \exp(it+ju+kv) = \cos(|\bold{w}|) + \sin(|\bold{w}|)\frac{\bold{w}}{|\bold{w}|}.\,</math>
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| : This map takes the 2-sphere of radius <math>R</math> inside the purely imaginary quaternions to <math>\{s\in S^3 \subset \bold{H}: \operatorname{Re}(s) = \cos(R)\} = </math> a 2-sphere of radius <math>\sin(R)</math> when <math>R\not\equiv 0\pmod{2\pi}</math>. Compare this to the first example above.
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| ===Properties===
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| *For all <math>X\in\mathfrak g</math>, the map <math>\gamma(t) = \exp(tX)</math> is the unique one-parameter subgroup of <math>G</math> whose [[tangent vector]] at the identity is <math>X</math>. It follows that:
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| **<math>\exp(t+s)X = (\exp tX)(\exp sX)\,</math>
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| **<math>\exp(-X) = (\exp X)^{-1}.\,</math>
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| *The exponential map <math>\exp\colon \mathfrak g \to G</math> is a [[smooth map]]. Its [[pushforward (differential)|derivative]] at the identity, <math>\exp_{*}\colon \mathfrak g \to \mathfrak g</math>, is the identity map (with the usual identifications). The exponential map, therefore, restricts to a [[diffeomorphism]] from some neighborhood of 0 in <math>\mathfrak g</math> to a neighborhood of 1 in <math>G</math>.
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| *The exponential map is not, however, a [[covering map]] in general – it is not a local diffeomorphism at all points. For example, so(3) to SO(3) is not a covering map; see also [[cut locus (Riemannian manifold)|cut locus]] on this failure.
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| *The image of the exponential map always lies in the [[identity component]] of <math>G</math>. When <math>G</math> is [[compact space|compact]], the exponential map is surjective onto the identity component.
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| *The image of the exponential map of the connected but non-compact group ''SL''<sub>2</sub>('''R''') is not the whole group. Its image consists of '''C'''-diagonalizable matrices with eigenvalues either positive or with module 1, and of non-diagonalizable trigonalizable matrices with eigenvalue 1.
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| *The map <math>\gamma(t) = \exp(tX)</math> is the [[integral curve]] through the identity of both the right- and left-invariant vector fields associated to <math>X</math>.
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| *The integral curve through <math>g\in G</math> of the left-invariant vector field <math>X^L</math> associated to <math>X</math> is given by <math>g \exp(t X)</math>. Likewise, the integral curve through <math>g</math> of the right-invariant vector field <math>X^R</math> is given by <math>\exp(t X) g</math>. It follows that the [[flow (geometry)|flow]]s <math>\xi^{L,R}</math> generated by the vector fields <math>X^{L,R}</math> are given by: {{unordered list|1=<math>\xi^L_t = R_{\exp tX}</math>|2=<math>\xi^R_t = L_{\exp tX}.</math>}} Since these flows are globally defined, every left- and right-invariant vector field on <math>G</math> is [[complete vector field|complete]].
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| *Let <math>\phi\colon G \to H</math> be a Lie group homomorphism and let <math>\phi_{*}</math> be its [[pushforward (differential)|derivative]] at the identity. Then the following diagram [[commutative diagram|commutes]]:
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| [[Image:ExponentialMap-01.png|center]]
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| *In particular, when applied to the [[adjoint representation of a Lie group|adjoint action]] of a group <math>G</math> we have
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| **<math>g(\exp X)g^{-1} = \exp(\mathrm{Ad}_gX)\,</math>
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| **<math>\mathrm{Ad}_{\exp X} = \exp(\mathrm{ad}_X).\,</math>
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| == Riemannian geometry ==
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| In [[Riemannian geometry]], an '''exponential map''' is a map from a subset of a [[tangent space]] T<sub>''p''</sub>''M'' of a [[Riemannian manifold]] (or [[pseudo-Riemannian manifold]]) ''M'' to ''M'' itself. The (pseudo) Riemannian metric determines a canonical affine connection, and the exponential map of the (pseudo) Riemannian manifold is given by the exponential map of this connection.
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| ===Properties===
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| Intuitively speaking, the exponential map takes a given tangent vector to the manifold, runs along the geodesic starting at that point and going in that direction, for a unit time. Since ''v'' corresponds to the velocity vector of the geodesic, the actual (Riemannian) distance traveled will be dependent on that. We can also reparametrize geodesics to be unit speed, so equivalently we can define exp<sub>''p''</sub>(''v'') = β(|''v''|) where β is the unit-speed geodesic (geodesic parameterized by arc length) going in the direction of ''v''. As we vary the tangent vector ''v'' we will get, when applying exp<sub>''p''</sub>, different points on ''M'' which are within some distance from the base point ''p''—this is perhaps one of the most concrete ways of demonstrating that the tangent space to a manifold is a kind of "linearization" of the manifold.
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| The [[Hopf–Rinow theorem]] asserts that it is possible to define the exponential map on the whole tangent space if and only if the manifold is complete as a [[metric space]] (which justifies the usual term '''geodesically complete''' for a manifold having an exponential map with this property). In particular, [[Compact space|compact]] manifolds are geodesically complete. However even if exp<sub>''p''</sub> is defined on the whole tangent space, it will in general not be a global [[diffeomorphism]]. However, its differential at the origin of the tangent space is the [[identity function|identity map]] and so, by the [[inverse function theorem]] we can find a neighborhood of the origin of T<sub>''p''</sub>''M'' on which the exponential map is an embedding (i.e., the exponential map is a local diffeomorphism). The radius of the largest ball about the origin in T<sub>''p''</sub>''M'' that can be mapped diffeomorphically via exp<sub>''p''</sub> is called the '''[[injectivity radius]]''' of ''M'' at ''p''. The [[Cut locus (Riemannian manifold)|cut locus]] of the exponential map is, roughly speaking, the set of all points where the exponential map fails to have a unique minimum.
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| An important property of the exponential map is the following [[Gauss's lemma (Riemannian geometry)|lemma of Gauss]] (yet another [[Gauss's lemma (disambiguation)|Gauss's lemma]]): given any tangent vector ''v'' in the domain of definition of exp<sub>''p''</sub>, and another vector ''w'' based at the tip of ''v'' (hence ''w'' is actually in the [[Double tangent bundle|double-tangent space]] T<sub>''v''</sub>(T<sub>''p''</sub>''M'')) and orthogonal to ''v'', remains orthogonal to ''v'' when pushed forward via the exponential map. This means, in particular, that the boundary sphere of a small ball about the origin in T<sub>''p''</sub>''M'' is orthogonal to the geodesics in ''M'' determined by those vectors (i.e., the geodesics are ''radial''). This motivates the definition of [[geodesic normal coordinates]] on a Riemannian manifold.
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| The exponential map is also useful in relating the [[Curvature of Riemannian manifolds|abstract definition of curvature]] to the more concrete realization of it originally conceived by Riemann himself—the [[sectional curvature]] is intuitively defined as the [[Gaussian curvature]] of some surface (i.e., a slicing of the manifold by a 2-dimensional submanifold) through the point ''p'' in consideration. Via the exponential map, it now can be precisely defined as the Gaussian curvature of a surface through ''p'' determined by the image under exp<sub>''p''</sub> of a 2-dimensional subspace of T<sub>''p''</sub>''M''.
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| == Relationships ==
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| In the case of Lie groups with a '''bi-invariant metric'''—a pseudo-Riemannian metric invariant under both left and right translation—the exponential maps of the pseudo-Riemannian structure are the same as the exponential maps of the Lie group. In general, Lie groups do not have a bi-invariant metric, though all connected semi-simple (or reductive) Lie groups do. The existence of a bi-invariant ''Riemannian'' metric is stronger than that of a pseudo-Riemannian metric, and implies that the Lie algebra is the Lie algebra of a compact Lie group; conversely, any compact (or abelian) Lie group has such a Riemannian metric.
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| Take the example that gives the "honest" exponential map. Consider the positive real numbers '''R'''<sup>+</sup>, a Lie group under the usual multiplication. Then each tangent space is just '''R'''. On each copy of '''R''' at the point ''y'', we introduce the modified inner product
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| :<math>\langle u,v\rangle_y = \frac{uv}{y^2}</math>
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| (multiplying them as usual real numbers but scaling by ''y''<sup>2</sup>). (This is what makes the metric left-invariant, for left multiplication by a factor will just pull out of the inner product, twice — canceling the square in the denominator).
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| Consider the point 1 ∈ '''R'''<sup>+</sup>, and ''x'' ∈ '''R''' an element of the tangent space at 1. The usual straight line emanating from 1, namely ''y''(''t'') = 1 + ''xt'' covers the same path as a geodesic, of course, except we have to reparametrize so as to get a curve with constant speed ("constant speed", remember, is not going to be the ordinary constant speed, because we're using this funny metric). To do this we reparametrize by arc length (the integral of the length of the tangent vector in the norm <math>|\cdot|_y</math> induced by the modified metric):
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| :<math>s(t) = \int_0^t |x|_{y(\tau)} d\tau = \int_0^t \frac{|x|}{1 + \tau x} d\tau = |x| \int_0^t \frac{d\tau}{1 + \tau x} = \frac{|x|}{x} \ln|1 + tx|</math>
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| and after inverting the function to obtain ''t'' as a function of ''s'', we substitute and get
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| :<math>y(s)=e^{sx/|x|}</math>
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| Now using the unit speed definition, we have
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| :<math>\exp_1(x)=y(|x|_1)=y(|x|)</math>,
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| giving the expected ''e''<sup>''x''</sup>.
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| The Riemannian distance defined by this is simply
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| :<math>\operatorname{dist}(a,b) = |\ln(b/a)|</math>,
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| a metric which should be familiar to anyone who has drawn graphs on [[graph paper|log paper]].
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| == See also ==
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| *[[List of exponential topics]]
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| == Notes ==
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| {{reflist}}
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| == References ==
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| * {{citation |first=Manfredo P. |last=do Carmo |title=Riemannian Geometry |publisher=Birkhäuser |year=1992 |isbn=0-8176-3490-8}}. See Chapter 3.
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| * {{citation |first1=Jeff |last1=Cheeger |first2=David G. |last2=Ebin |title=Comparison Theorems in Riemannian Geometry |publisher=Elsevier |year=1975}}. See Chapter 1, Sections 2 and 3.
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| * {{springer|title=Exponential mapping|id=p/e036930}}
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| * {{Citation | last1=Helgason | first1=Sigurdur | title=Differential geometry, Lie groups, and symmetric spaces | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Graduate Studies in Mathematics | isbn=978-0-8218-2848-9 | mr=1834454 | year=2001 | volume=34}}.
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| * {{citation | last1=Kobayashi |first1=Shoshichi |last2=Nomizu |first2=Katsumi | title = [[Foundations of Differential Geometry]] |volume=Vol. 1 | publisher=Wiley-Interscience | year=1996 |edition=New |isbn=0-471-15733-3}}.
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| {{Chaos theory}}
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| {{DEFAULTSORT:Exponential Map}}
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| [[Category:Exponentials]]
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| [[Category:Lie groups]]
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| [[Category:Riemannian geometry]]
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