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Tangent space

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Assignment of vector fields to manifolds

Inmathematics, thetangent space of amanifold is a generalization oftangent lines to curves intwo-dimensional space andtangent planes to surfaces inthree-dimensional space in higher dimensions. In the context of physics, the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold.

Informal description

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A pictorial representation of the tangent space of a single pointP(r,θ,ϕ){\displaystyle P(r,\theta ,\phi )} on asphere. A vector in this tangent space represents a possible velocity (of something moving on the sphere) atP{\displaystyle P}. After moving in that direction to a nearby point, the velocity would then be given by a vector in the tangent space of that point—a different tangent space that is not shown.

Indifferential geometry, one can attach to every pointx{\displaystyle x} of adifferentiable manifold atangent space—a realvector space that intuitively contains the possible directions in which one can tangentially pass throughx{\displaystyle x}. The elements of the tangent space atx{\displaystyle x} are called thetangent vectors atx{\displaystyle x}. This is a generalization of the notion of avector, based at a given initial point, in aEuclidean space. Thedimension of the tangent space at every point of aconnected manifold is the same as that of themanifold itself.

For example, if the given manifold is a2{\displaystyle 2}-sphere, then one can picture the tangent space at a point as the plane that touches the sphere at that point and isperpendicular to the sphere's radius through the point. More generally, if a given manifold is thought of as anembeddedsubmanifold ofEuclidean space, then one can picture a tangent space in this literal fashion. This was the traditional approach toward definingparallel transport. Many authors indifferential geometry andgeneral relativity use it.[1][2] More strictly, this defines an affine tangent space, which is distinct from the space of tangent vectors described by modern terminology.

Inalgebraic geometry, in contrast, there is an intrinsic definition of thetangent space at a point of analgebraic varietyV{\displaystyle V} that gives a vector space with dimension at least that ofV{\displaystyle V} itself. The pointsp{\displaystyle p} at which the dimension of the tangent space is exactly that ofV{\displaystyle V} are callednon-singular points; the others are calledsingular points. For example, a curve that crosses itself does not have a unique tangent line at that point. The singular points ofV{\displaystyle V} are those where the "test to be a manifold" fails. SeeZariski tangent space.

Once the tangent spaces of a manifold have been introduced, one can definevector fields, which are abstractions of the velocity field of particles moving in space. A vector field attaches to every point of the manifold a vector from the tangent space at that point, in a smooth manner. Such a vector field serves to define a generalizedordinary differential equation on a manifold: A solution to such a differential equation is a differentiablecurve on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field.

All the tangent spaces of a manifold may be "glued together" to form a new differentiable manifold with twice the dimension of the original manifold, called thetangent bundle of the manifold.

Formal definitions

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The informal description above relies on a manifold's ability to be embedded into an ambient vector spaceRm{\displaystyle \mathbb {R} ^{m}} so that the tangent vectors can "stick out" of the manifold into the ambient space. However, it is more convenient to define the notion of a tangent space based solely on the manifold itself.[3]

There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via the velocity of curves is intuitively the simplest, it is also the most cumbersome to work with. More elegant and abstract approaches are described below.

Definition via tangent curves

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In the embedded-manifold picture, a tangent vector at a pointx{\displaystyle x} is thought of as thevelocity of acurve passing through the pointx{\displaystyle x}. We can therefore define a tangent vector as an equivalence class of curves passing throughx{\displaystyle x} while being tangent to each other atx{\displaystyle x}.

Suppose thatM{\displaystyle M} is aCk{\displaystyle C^{k}}differentiable manifold (withsmoothnessk≥1{\displaystyle k\geq 1}) and thatx∈M{\displaystyle x\in M}. Pick acoordinate chartφ:U→Rn{\displaystyle \varphi :U\to \mathbb {R} ^{n}}, whereU{\displaystyle U} is anopen subset ofM{\displaystyle M} containingx{\displaystyle x}. Suppose further that two curvesγ1,γ2:(−1,1)→M{\displaystyle \gamma _{1},\gamma _{2}:(-1,1)\to M} withγ1(0)=x=γ2(0){\displaystyle {\gamma _{1}}(0)=x={\gamma _{2}}(0)} are given such that bothφ∘γ1,φ∘γ2:(−1,1)→Rn{\displaystyle \varphi \circ \gamma _{1},\varphi \circ \gamma _{2}:(-1,1)\to \mathbb {R} ^{n}} are differentiable in the ordinary sense (we call thesedifferentiable curves initialized atx{\displaystyle x}). Thenγ1{\displaystyle \gamma _{1}} andγ2{\displaystyle \gamma _{2}} are said to beequivalent at0{\displaystyle 0} if and only if the derivatives ofφ∘γ1{\displaystyle \varphi \circ \gamma _{1}} andφ∘γ2{\displaystyle \varphi \circ \gamma _{2}} at0{\displaystyle 0} coincide. This defines anequivalence relation on the set of all differentiable curves initialized atx{\displaystyle x}, andequivalence classes of such curves are known astangent vectors ofM{\displaystyle M} atx{\displaystyle x}. The equivalence class of any such curveγ{\displaystyle \gamma } is denoted byγ′(0){\displaystyle \gamma '(0)}. Thetangent space ofM{\displaystyle M} atx{\displaystyle x}, denoted byTxM{\displaystyle T_{x}M}, is then defined as the set of all tangent vectors atx{\displaystyle x}; it does not depend on the choice of coordinate chartφ:U→Rn{\displaystyle \varphi :U\to \mathbb {R} ^{n}}.

The tangent spaceTxM{\displaystyle T_{x}M} and a tangent vectorv∈TxM{\displaystyle v\in T_{x}M}, along a curve traveling throughx∈M{\displaystyle x\in M}.

To define vector-space operations onTxM{\displaystyle T_{x}M}, we use a chartφ:U→Rn{\displaystyle \varphi :U\to \mathbb {R} ^{n}} and define amapdφx:TxM→Rn{\displaystyle \mathrm {d} {\varphi }_{x}:T_{x}M\to \mathbb {R} ^{n}} bydφx(γ′(0)):=d(φ∘γ)dt(0),{\textstyle {\mathrm {d} {\varphi }_{x}}(\gamma '(0)):={\frac {\mathrm {d} (\varphi \circ \gamma )}{\mathrm {d} {t}}}(0),} whereγ∈γ′(0){\displaystyle \gamma \in \gamma '(0)}. The mapdφx{\displaystyle \mathrm {d} {\varphi }_{x}} turns out to bebijective and may be used to transfer the vector-space operations onRn{\displaystyle \mathbb {R} ^{n}} over toTxM{\displaystyle T_{x}M}, thus turning the latter set into ann{\displaystyle n}-dimensional real vector space. Again, one needs to check that this construction does not depend on the particular chartφ:U→Rn{\displaystyle \varphi :U\to \mathbb {R} ^{n}} and the curveγ{\displaystyle \gamma } being used, and in fact it does not.

Definition via derivations

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Suppose now thatM{\displaystyle M} is aC∞{\displaystyle C^{\infty }} manifold. A real-valued functionf:M→R{\displaystyle f:M\to \mathbb {R} } is said to belong toC∞(M){\displaystyle {C^{\infty }}(M)} if and only if for every coordinate chartφ:U→Rn{\displaystyle \varphi :U\to \mathbb {R} ^{n}}, the mapf∘φ−1:φ[U]⊆Rn→R{\displaystyle f\circ \varphi ^{-1}:\varphi [U]\subseteq \mathbb {R} ^{n}\to \mathbb {R} } is infinitely differentiable. Note thatC∞(M){\displaystyle {C^{\infty }}(M)} is a realassociative algebra with respect to thepointwise product and sum of functions and scalar multiplication.

Aderivation atx∈M{\displaystyle x\in M} is defined as alinear mapD:C∞(M)→R{\displaystyle D:{C^{\infty }}(M)\to \mathbb {R} } that satisfies the Leibniz identity∀f,g∈C∞(M):D(fg)=D(f)⋅g(x)+f(x)⋅D(g),{\displaystyle \forall f,g\in {C^{\infty }}(M):\qquad D(fg)=D(f)\cdot g(x)+f(x)\cdot D(g),}which is modeled on theproduct rule of calculus.

(For every identically constant functionf=const,{\displaystyle f={\text{const}},} it follows thatD(f)=0{\displaystyle D(f)=0}).

DenoteTxM{\displaystyle T_{x}M} the set of all derivations atx.{\displaystyle x.} Setting

turnsTxM{\displaystyle T_{x}M} into a vector space.

Generalizations

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Generalizations of this definition are possible, for instance, tocomplex manifolds andalgebraic varieties. However, instead of examining derivationsD{\displaystyle D} from the full algebra of functions, one must instead work at the level ofgerms of functions. The reason for this is that thestructure sheaf may not befine for such structures. For example, letX{\displaystyle X} be an algebraic variety withstructure sheafOX{\displaystyle {\mathcal {O}}_{X}}. Then theZariski tangent space at a pointp∈X{\displaystyle p\in X} is the collection of allk{\displaystyle \mathbb {k} }-derivationsD:OX,p→k{\displaystyle D:{\mathcal {O}}_{X,p}\to \mathbb {k} }, wherek{\displaystyle \mathbb {k} } is theground field andOX,p{\displaystyle {\mathcal {O}}_{X,p}} is thestalk ofOX{\displaystyle {\mathcal {O}}_{X}} atp{\displaystyle p}.

Equivalence of the definitions

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Forx∈M{\displaystyle x\in M} and a differentiable curveγ:(−1,1)→M{\displaystyle \gamma :(-1,1)\to M} such thatγ(0)=x,{\displaystyle \gamma (0)=x,} defineDγ(f):=(f∘γ)′(0){\displaystyle {D_{\gamma }}(f):=(f\circ \gamma )'(0)} (where the derivative is taken in the ordinary sense becausef∘γ{\displaystyle f\circ \gamma } is a function from(−1,1){\displaystyle (-1,1)} toR{\displaystyle \mathbb {R} }). One can ascertain thatDγ(f){\displaystyle D_{\gamma }(f)} is a derivation at the pointx,{\displaystyle x,} and that equivalent curves yield the same derivation. Thus, for an equivalence classγ′(0),{\displaystyle \gamma '(0),} we can defineDγ′(0)(f):=(f∘γ)′(0),{\displaystyle {D_{\gamma '(0)}}(f):=(f\circ \gamma )'(0),} where the curveγ∈γ′(0){\displaystyle \gamma \in \gamma '(0)} has been chosen arbitrarily. The mapγ′(0)↦Dγ′(0){\displaystyle \gamma '(0)\mapsto D_{\gamma '(0)}} is a vector space isomorphism between the space of the equivalence classesγ′(0){\displaystyle \gamma '(0)} and the space of derivations at the pointx.{\displaystyle x.}

Definition via cotangent spaces

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Again, we start with aC∞{\displaystyle C^{\infty }} manifoldM{\displaystyle M} and a pointx∈M{\displaystyle x\in M}. Consider theidealI{\displaystyle I} ofC∞(M){\displaystyle C^{\infty }(M)} that consists of all smooth functionsf{\displaystyle f} vanishing atx{\displaystyle x}, i.e.,f(x)=0{\displaystyle f(x)=0}. ThenI{\displaystyle I} andI2{\displaystyle I^{2}} are both real vector spaces, and thequotient spaceI/I2{\displaystyle I/I^{2}} can be shown to be isomorphic to thecotangent spaceTx∗M{\displaystyle T_{x}^{*}M} through the use ofTaylor's theorem. The tangent spaceTxM{\displaystyle T_{x}M} may then be defined as thedual space ofI/I2{\displaystyle I/I^{2}}.

While this definition is the most abstract, it is also the one that is most easily transferable to other settings, for instance, to thevarieties considered inalgebraic geometry.

IfD{\displaystyle D} is a derivation atx{\displaystyle x}, thenD(f)=0{\displaystyle D(f)=0} for everyf∈I2{\displaystyle f\in I^{2}}, which means thatD{\displaystyle D} gives rise to a linear mapI/I2→R{\displaystyle I/I^{2}\to \mathbb {R} }. Conversely, ifr:I/I2→R{\displaystyle r:I/I^{2}\to \mathbb {R} } is a linear map, thenD(f):=r((f−f(x))+I2){\displaystyle D(f):=r\left((f-f(x))+I^{2}\right)} defines a derivation atx{\displaystyle x}. This yields an equivalence between tangent spaces defined via derivations and tangent spaces defined via cotangent spaces.

Properties

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IfM{\displaystyle M} is an open subset ofRn{\displaystyle \mathbb {R} ^{n}}, thenM{\displaystyle M} is aC∞{\displaystyle C^{\infty }} manifold in a natural manner (take coordinate charts to beidentity maps on open subsets ofRn{\displaystyle \mathbb {R} ^{n}}), and the tangent spaces are all naturally identified withRn{\displaystyle \mathbb {R} ^{n}}.

Tangent vectors as directional derivatives

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Another way to think about tangent vectors is asdirectional derivatives. Given a vectorv{\displaystyle v} inRn{\displaystyle \mathbb {R} ^{n}}, one defines the corresponding directional derivative at a pointx∈Rn{\displaystyle x\in \mathbb {R} ^{n}} by

∀f∈C∞(Rn):(Dvf)(x):=ddt[f(x+tv)]|t=0=∑i=1nvi∂f∂xi(x).{\displaystyle \forall f\in {C^{\infty }}(\mathbb {R} ^{n}):\qquad (D_{v}f)(x):=\left.{\frac {\mathrm {d} }{\mathrm {d} {t}}}[f(x+tv)]\right|_{t=0}=\sum _{i=1}^{n}v^{i}{\frac {\partial f}{\partial x^{i}}}(x).}

This map is naturally a derivation atx{\displaystyle x}. Furthermore, every derivation at a point inRn{\displaystyle \mathbb {R} ^{n}} is of this form. Hence, there is a one-to-one correspondence between vectors (thought of as tangent vectors at a point) and derivations at a point.

As tangent vectors to a general manifold at a point can be defined as derivations at that point, it is natural to think of them as directional derivatives. Specifically, ifv{\displaystyle v} is a tangent vector toM{\displaystyle M} at a pointx{\displaystyle x} (thought of as a derivation), then define the directional derivativeDv{\displaystyle D_{v}} in the directionv{\displaystyle v} by

∀f∈C∞(M):Dv(f):=v(f).{\displaystyle \forall f\in {C^{\infty }}(M):\qquad {D_{v}}(f):=v(f).}

If we think ofv{\displaystyle v} as the initial velocity of a differentiable curveγ{\displaystyle \gamma } initialized atx{\displaystyle x}, i.e.,v=γ′(0){\displaystyle v=\gamma '(0)}, then instead, defineDv{\displaystyle D_{v}} by

∀f∈C∞(M):Dv(f):=(f∘γ)′(0).{\displaystyle \forall f\in {C^{\infty }}(M):\qquad {D_{v}}(f):=(f\circ \gamma )'(0).}

Basis of the tangent space at a point

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For aC∞{\displaystyle C^{\infty }} manifoldM{\displaystyle M}, if a chartφ=(x1,…,xn):U→Rn{\displaystyle \varphi =(x^{1},\ldots ,x^{n}):U\to \mathbb {R} ^{n}} is given withp∈U{\displaystyle p\in U}, then one can define an ordered basis{∂∂x1|p,…,∂∂xn|p}{\textstyle \left\{\left.{\frac {\partial }{\partial x^{1}}}\right|_{p},\dots ,\left.{\frac {\partial }{\partial x^{n}}}\right|_{p}\right\}} ofTpM{\displaystyle T_{p}M} by

∀i∈{1,…,n}, ∀f∈C∞(M):∂∂xi|p(f):=(∂∂xi(f∘φ−1))(φ(p)).{\displaystyle \forall i\in \{1,\ldots ,n\},~\forall f\in {C^{\infty }}(M):\qquad {\left.{\frac {\partial }{\partial x^{i}}}\right|_{p}}(f):=\left({\frac {\partial }{\partial x^{i}}}{\Big (}f\circ \varphi ^{-1}{\Big )}\right){\Big (}\varphi (p){\Big )}.}

Then for every tangent vectorv∈TpM{\displaystyle v\in T_{p}M}, one has

v=∑i=1nvi∂∂xi|p.{\displaystyle v=\sum _{i=1}^{n}v^{i}\left.{\frac {\partial }{\partial x^{i}}}\right|_{p}.}

This formula therefore expressesv{\displaystyle v} as a linear combination of the basis tangent vectors∂∂xi|p∈TpM{\textstyle \left.{\frac {\partial }{\partial x^{i}}}\right|_{p}\in T_{p}M} defined by the coordinate chartφ:U→Rn{\displaystyle \varphi :U\to \mathbb {R} ^{n}}.[4]

The derivative of a map

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Main article:Pushforward (differential)

Every smooth (or differentiable) mapφ:M→N{\displaystyle \varphi :M\to N} between smooth (or differentiable) manifolds induces naturallinear maps between their corresponding tangent spaces:

dφx:TxM→Tφ(x)N.{\displaystyle \mathrm {d} {\varphi }_{x}:T_{x}M\to T_{\varphi (x)}N.}

If the tangent space is defined via differentiable curves, then this map is defined by

dφx(γ′(0)):=(φ∘γ)′(0).{\displaystyle {\mathrm {d} {\varphi }_{x}}(\gamma '(0)):=(\varphi \circ \gamma )'(0).}

If, instead, the tangent space is defined via derivations, then this map is defined by

[dφx(D)](f):=D(f∘φ).{\displaystyle [\mathrm {d} {\varphi }_{x}(D)](f):=D(f\circ \varphi ).}

The linear mapdφx{\displaystyle \mathrm {d} {\varphi }_{x}} is called variously thederivative,total derivative,differential, orpushforward ofφ{\displaystyle \varphi } atx{\displaystyle x}. It is frequently expressed using a variety of other notations:

Dφx,(φ∗)x,φ′(x).{\displaystyle D\varphi _{x},\qquad (\varphi _{*})_{x},\qquad \varphi '(x).}

In a sense, the derivative is the best linear approximation toφ{\displaystyle \varphi } nearx{\displaystyle x}. Note that whenN=R{\displaystyle N=\mathbb {R} }, then the mapdφx:TxM→R{\displaystyle \mathrm {d} {\varphi }_{x}:T_{x}M\to \mathbb {R} } coincides with the usual notion of thedifferential of the functionφ{\displaystyle \varphi }. Inlocal coordinates the derivative ofφ{\displaystyle \varphi } is given by theJacobian.

An important result regarding the derivative map is the following:

Theorem—Ifφ:M→N{\displaystyle \varphi :M\to N} is alocal diffeomorphism atx{\displaystyle x} inM{\displaystyle M}, thendφx:TxM→Tφ(x)N{\displaystyle \mathrm {d} {\varphi }_{x}:T_{x}M\to T_{\varphi (x)}N} is a linearisomorphism. Conversely, ifφ:M→N{\displaystyle \varphi :M\to N} is continuously differentiable anddφx{\displaystyle \mathrm {d} {\varphi }_{x}} is an isomorphism, then there is anopen neighborhoodU{\displaystyle U} ofx{\displaystyle x} such thatφ{\displaystyle \varphi } mapsU{\displaystyle U} diffeomorphically onto its image.

This is a generalization of theinverse function theorem to maps between manifolds.

See also

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Notes

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  1. ^do Carmo, Manfredo P. (1976).Differential Geometry of Curves and Surfaces. Prentice-Hall.:
  2. ^Dirac, Paul A. M. (1996) [1975].General Theory of Relativity. Princeton University Press.ISBN 0-691-01146-X.
  3. ^Chris J. Isham (1 January 2002).Modern Differential Geometry for Physicists. Allied Publishers. pp. 70–72.ISBN 978-81-7764-316-9.
  4. ^Lerman, Eugene."An Introduction to Differential Geometry"(PDF). p. 12. Archived fromthe original(PDF) on 2023-06-08. Retrieved2021-04-09.

References

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External links

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