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In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other.

Informal description

A pictorial representation of the tangent space of a single point x{displaystyle x} on a sphere. A vector in this tangent space represents a possible velocity at x{displaystyle x}. 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.

In differential geometry, one can attach to every point x{displaystyle x} of a differentiable manifold a tangent space—a real vector space that intuitively contains the possible directions in which one can tangentially pass through x{displaystyle x}. The elements of the tangent space at x{displaystyle x} are called the tangent vectors at x{displaystyle x}. This is a generalization of the notion of a bound vector in a Euclidean space. The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself.

For example, if the given manifold is a 2{displaystyle 2}-sphere, then one can picture the tangent space at a point as the plane that touches the sphere at that point and is perpendicular to the sphere's radius through the point. More generally, if a given manifold is thought of as an embedded submanifold of Euclidean space, then one can picture a tangent space in this literal fashion. This was the traditional approach toward defining parallel transport and was used by Dirac.^[1] More strictly, this defines an affine tangent space, which is distinct from the space of tangent vectors described by modern terminology.

In algebraic geometry, in contrast, there is an intrinsic definition of the tangent space at a point of an algebraic variety V{displaystyle V} that gives a vector space with dimension at least that of V{displaystyle V} itself. The points p{displaystyle p} at which the dimension of the tangent space is exactly that of V{displaystyle V} are called non-singular points; the others are called singular points. For example, a curve that crosses itself does not have a unique tangent line at that point. The singular points of V{displaystyle V} are those where the ‘test to be a manifold’ fails. See Zariski tangent space.

Once the tangent spaces of a manifold have been introduced, one can define vector 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 generalized ordinary differential equation on a manifold: A solution to such a differential equation is a differentiable curve 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 the tangent bundle of the manifold.

Formal definitions

The informal description above relies on a manifold's ability to be embedded into an ambient vector space Rm{displaystyle mathbf {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.^[2]

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 as the velocity of curves

In the embedded-manifold picture, a tangent vector at a point x{displaystyle x} is thought of as the velocity of a curve passing through the point x{displaystyle x}. We can therefore define a tangent vector as an equivalence class of curves passing through x{displaystyle x} while being tangent to each other at x{displaystyle x}.

Suppose that M{displaystyle M} is a Ck{displaystyle C^{k}} manifold (k≥1{displaystyle kgeq 1}) and that x∈M{displaystyle xin M}. Pick a coordinate chart φ:U→Rn{displaystyle varphi :Uto mathbf {R} ^{n}}, where U{displaystyle U} is an open subset of M{displaystyle M} containing x{displaystyle x}. Suppose further that two curves γ1,γ2:(−1,1)→M{displaystyle gamma _{1},gamma _{2}:(-1,1)to M} with γ1(0)=x=γ2(0){displaystyle {gamma _{1}}(0)=x={gamma _{2}}(0)} are given such that both φ∘γ1,φ∘γ2:(−1,1)→Rn{displaystyle varphi circ gamma _{1},varphi circ gamma _{2}:(-1,1)to mathbf {R} ^{n}} are differentiable in the ordinary sense (we call these differentiable curves initialized at x{displaystyle x}). Then γ1{displaystyle gamma _{1}} and γ2{displaystyle gamma _{2}} are said to be equivalent at 0{displaystyle 0} if and only if the derivatives of φ∘γ1{displaystyle varphi circ gamma _{1}} and φ∘γ2{displaystyle varphi circ gamma _{2}} at 0{displaystyle 0} coincide. This defines an equivalence relation on the set of all differentiable curves initialized at x{displaystyle x}, and equivalence classes of such curves are known as tangent vectors of M{displaystyle M} at x{displaystyle x}. The equivalence class of any such curve γ{displaystyle gamma } is denoted by γ′(0){displaystyle gamma '(0)}. The tangent space of M{displaystyle M} at x{displaystyle x}, denoted by TxM{displaystyle T_{x}M}, is then defined as the set of all tangent vectors at x{displaystyle x}; it does not depend on the choice of coordinate chart φ:U→Rn{displaystyle varphi :Uto mathbf {R} ^{n}}.

The tangent space TxM{displaystyle T_{x}M} and a tangent vector v∈TxM{displaystyle vin T_{x}M}, along a curve traveling through x∈M{displaystyle xin M}.

To define vector-space operations on TxM{displaystyle T_{x}M}, we use a chart φ:U→Rn{displaystyle varphi :Uto mathbf {R} ^{n}} and define a map dφx:TxM→Rn{displaystyle mathrm {d} {varphi }_{x}:T_{x}Mto mathbf {R} ^{n}} by dφx(γ′(0)) =df ddt[(φ∘γ)(t)]|t=0{displaystyle {mathrm {d} {varphi }_{x}}(gamma '(0))~{stackrel {text{df}}{=}}~left.{frac {mathrm {d} }{mathrm {d} {t}}}[(varphi circ gamma )(t)]right|_{t=0}}. This map turns out to be bijective and may be used to transfer the vector-space operations on Rn{displaystyle mathbf {R} ^{n}} over to TxM{displaystyle T_{x}M}, thus turning the latter set into an n{displaystyle n}-dimensional real vector space. Again, one needs to check that this construction does not depend on the particular chart φ:U→Rn{displaystyle varphi :Uto mathbf {R} ^{n}} being used, and in fact it does not.

Definition via derivations

Suppose now that M{displaystyle M} is a C∞{displaystyle C^{infty }} manifold. A real-valued function f:M→R{displaystyle f:Mto mathbf {R} } is said to belong to C∞(M){displaystyle {C^{infty }}(M)} if and only if for every coordinate chart φ:U→Rn{displaystyle varphi :Uto mathbf {R} ^{n}}, the map f∘φ−1:φ[U]⊆Rn→R{displaystyle fcirc varphi ^{-1}:varphi [U]subseteq mathbf {R} ^{n}to mathbf {R} } is infinitely differentiable. Note that C∞(M){displaystyle {C^{infty }}(M)} is a real associative algebra with respect to the pointwise product and sum of functions and scalar multiplication.

Pick a point x∈M{displaystyle xin M}. A derivation at x{displaystyle x} is defined as a linear map D:C∞(M)→R{displaystyle D:{C^{infty }}(M)to mathbf {R} } that satisfies the Leibniz identity

∀f,g∈C∞(M):D(fg)=D(f)⋅g(x)+f(x)⋅D(g),{displaystyle forall f,gin {C^{infty }}(M):qquad D(fg)=D(f)cdot g(x)+f(x)cdot D(g),}

which is modeled on the product rule of calculus.

If we define addition and scalar multiplication on the set of derivations at x{displaystyle x} by

• 
  (D1+D2)(f) =df D1(f)+D2(f){displaystyle (D_{1}+D_{2})(f)~{stackrel {text{df}}{=}}~{D_{1}}(f)+{D_{2}}(f)} and


• 
  (λ⋅D)(f) =df λ⋅D(f){displaystyle (lambda cdot D)(f)~{stackrel {text{df}}{=}}~lambda cdot D(f)},

then we obtain a real vector space, which we define as the tangent space TxM{displaystyle T_{x}M} of M{displaystyle M} at x{displaystyle x}.

The relation between derivations at a point x{displaystyle x} and tangent vectors at x{displaystyle x} is as follows: If γ:(−1,1)→M{displaystyle gamma :(-1,1)to M} is a differentiable curve initialized at x{displaystyle x}, then the corresponding derivation Dγ{displaystyle D_{gamma }} at x{displaystyle x} is defined by Dγ(f) =df (f∘γ)′(0){displaystyle {D_{gamma }}(f)~{stackrel {text{df}}{=}}~(fcirc gamma )'(0)} (where the derivative is taken in the ordinary sense because f∘γ{displaystyle fcirc gamma } is a function from (−1,1){displaystyle (-1,1)} to R{displaystyle mathbf {R} }).

Generalizations of this definition are possible, for instance, to complex manifolds and algebraic varieties. However, instead of examining derivations D{displaystyle D} from the full algebra of functions, one must instead work at the level of germs of functions. The reason for this is that the structure sheaf may not be fine for such structures. For example, let X{displaystyle X} be an algebraic variety with structure sheaf OX{displaystyle {mathcal {O}}_{X}}. Then the Zariski tangent space at a point p∈X{displaystyle pin X} is the collection of all k{displaystyle mathbb {k} }-derivations D:OX,p→k{displaystyle D:{mathcal {O}}_{X,p}to mathbb {k} }, where k{displaystyle mathbb {k} } is the ground field and OX,p{displaystyle {mathcal {O}}_{X,p}} is the stalk of OX{displaystyle {mathcal {O}}_{X}} at p{displaystyle p}.

Definition via cotangent spaces

Again, we start with a C∞{displaystyle C^{infty }} manifold M{displaystyle M} and a point x∈M{displaystyle xin M}. Consider the ideal I{displaystyle I} of C∞(M){displaystyle C^{infty }(M)} that consists of all smooth functions f{displaystyle f} vanishing at x{displaystyle x}, i.e., f(x)=0{displaystyle f(x)=0}. Then I{displaystyle I} and I2{displaystyle I^{2}} are real vector spaces, and TxM{displaystyle T_{x}M} may be defined as the dual space of the quotient space I/I2{displaystyle I/I^{2}}. This latter quotient space is also known as the cotangent space of M{displaystyle M} at x{displaystyle x}.

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

If D{displaystyle D} is a derivation at x{displaystyle x}, then D(f)=0{displaystyle D(f)=0} for every f∈I2{displaystyle fin I^{2}}, which means that D{displaystyle D} gives rise to a linear map I/I2→R{displaystyle I/I^{2}to mathbf {R} }. Conversely, if r:I/I2→R{displaystyle r:I/I^{2}to mathbf {R} } is a linear map, then D(f) =def r((f−f(x))+I2){displaystyle D(f)~{stackrel {text{def}}{=}}~rleft((f-f(x))+I^{2}right)} defines a derivation at x{displaystyle x}. This yields an equivalence between tangent spaces defined via derivations and tangent spaces defined via cotangent spaces.

Properties

If M{displaystyle M} is an open subset of Rn{displaystyle mathbf {R} ^{n}}, then M{displaystyle M} is a C∞{displaystyle C^{infty }} manifold in a natural manner (take coordinate charts to be identity maps on open subsets of Rn{displaystyle mathbf {R} ^{n}}), and the tangent spaces are all naturally identified with Rn{displaystyle mathbf {R} ^{n}}.

Tangent vectors as directional derivatives

Another way to think about tangent vectors is as directional derivatives. Given a vector v{displaystyle v} in Rn{displaystyle mathbf {R} ^{n}}, one defines the corresponding directional derivative at a point x∈Rn{displaystyle xin mathbf {R} ^{n}} by

∀f∈C∞(Rn):(Dvf)(x) =df ddt[f(x+tv)]|t=0=∑i=1nvi∂f∂xi(x).{displaystyle forall fin {C^{infty }}(mathbb {R} ^{n}):qquad (D_{v}f)(x)~{stackrel {text{df}}{=}}~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 at x{displaystyle x}. Furthermore, every derivation at a point in Rn{displaystyle mathbf {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, if v{displaystyle v} is a tangent vector to M{displaystyle M} at a point x{displaystyle x} (thought of as a derivation), then define the directional derivative Dv{displaystyle D_{v}} in the direction v{displaystyle v} by

∀f∈C∞(M):Dv(f) =df v(f).{displaystyle forall fin {C^{infty }}(M):qquad {D_{v}}(f)~{stackrel {text{df}}{=}}~v(f).}

If we think of v{displaystyle v} as the initial velocity of a differentiable curve γ{displaystyle gamma } initialized at x{displaystyle x}, i.e., v=γ′(0){displaystyle v=gamma '(0)}, then instead, define Dv{displaystyle D_{v}} by

∀f∈C∞(M):Dv(f) =df (f∘γ)′(0).{displaystyle forall fin {C^{infty }}(M):qquad {D_{v}}(f)~{stackrel {text{df}}{=}}~(fcirc gamma )'(0).}

Basis of the tangent space at a point

For a C∞{displaystyle C^{infty }} manifold M{displaystyle M}, if a chart φ=(x1,…,xn):U→Rn{displaystyle varphi =(x^{1},ldots ,x^{n}):Uto mathbf {R} ^{n}} is given with p∈U{displaystyle pin U}, then one can define an ordered basis ((∂∂xi)p)i=1n{displaystyle left(left({frac {partial }{partial x^{i}}}right)_{p}right)_{i=1}^{n}} of TpM{displaystyle T_{p}M} by

∀i∈{1,…,n}, ∀f∈C∞(M):(∂∂xi)p(f) =df (∂i(f∘φ−1))(φ(p)).{displaystyle forall iin {1,ldots ,n},~forall fin {C^{infty }}(M):qquad {left({frac {partial }{partial x^{i}}}right)_{p}}(f)~{stackrel {text{df}}{=}}~({partial _{i}}(fcirc varphi ^{-1}))(varphi (p)).}

Then for every tangent vector v∈TpM{displaystyle vin T_{p}M}, one has

v=∑i=1nv(xi)⋅(∂∂xi)p.{displaystyle v=sum _{i=1}^{n}v(x^{i})cdot left({frac {partial }{partial x^{i}}}right)_{p}.}

This formula therefore expresses v{displaystyle v} as a linear combination of the basis tangent vectors (∂∂xi)p∈TpM{displaystyle left({frac {partial }{partial x^{i}}}right)_{p}in T_{p}M} defined by the coordinate chart φ:U→Rn{displaystyle varphi :Uto mathbf {R} ^{n}}.^[3]

The derivative of a map

Every smooth (or differentiable) map φ:M→N{displaystyle varphi :Mto N} between smooth (or differentiable) manifolds induces natural linear maps between their corresponding tangent spaces:

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

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

dφx(γ′(0)) =df (φ∘γ)′(0).{displaystyle {mathrm {d} {varphi }_{x}}(gamma '(0))~{stackrel {text{df}}{=}}~(varphi circ gamma )'(0).}

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

[dφx(X)](f) =df X(f∘φ).{displaystyle [mathrm {d} {varphi }_{x}(X)](f)~{stackrel {text{df}}{=}}~X(fcirc varphi ).}

The linear map dφx{displaystyle mathrm {d} {varphi }_{x}} is called variously the derivative, total derivative, differential, or pushforward of φ{displaystyle varphi } at x{displaystyle x}. It is frequently expressed using a variety of other notations:

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

In a sense, the derivative is the best linear approximation to φ{displaystyle varphi } near x{displaystyle x}. Note that when N=R{displaystyle N=mathbf {R} }, then the map dφx:TxM→R{displaystyle mathrm {d} {varphi }_{x}:T_{x}Mto mathbf {R} } coincides with the usual notion of the differential of the function φ{displaystyle varphi }. In local coordinates the derivative of φ{displaystyle varphi } is given by the Jacobian.

An important result regarding the derivative map is the following:

Theorem. If φ:M→N{displaystyle varphi :Mto N} is a local diffeomorphism at x{displaystyle x} in M{displaystyle M}, then dφx:TxM→Tφ(x)N{displaystyle mathrm {d} {varphi }_{x}:T_{x}Mto T_{varphi (x)}N} is a linear isomorphism. Conversely, if dφx{displaystyle mathrm {d} {varphi }_{x}} is an isomorphism, then there is an open neighborhood U{displaystyle U} of x{displaystyle x} such that φ{displaystyle varphi } maps U{displaystyle U} diffeomorphically onto its image.

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

See also

• Exponential map


• Vector space


• Differential geometry of curves


• Coordinate-induced basis


• Cotangent space

References

• 
  Lee, Jeffrey M. (2009), Manifolds and Differential Geometry, Graduate Studies in Mathematics, Vol. 107, Providence: American Mathematical Society.


• 
  Michor, Peter W. (2008), Topics in Differential Geometry, Graduate Studies in Mathematics, Vol. 93, Providence: American Mathematical Society.


• 
  Spivak, Michael (1965), Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus, W. A. Benjamin, Inc., ISBN 978-0-8053-9021-6.

External links

• 
  Tangent Planes at MathWorld