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Submanifold Tangent Space


The tangent plane to a surface at a point p is the tangent space at p (after translating to the origin). The elements of the tangent space are called tangent vectors, and they are closed under addition and scalar multiplication. In particular, the tangent space is a vector space.

Any submanifold of Euclidean space, and more generally any submanifold of an abstract manifold, has a tangent space at each point. The collection of tangent spaces TM_p to M forms the tangent bundle TM= union _(p in M)(p,TM_p). A vector field assigns to every point p a tangent vector in the tangent space at p.

There are two ways of defining a submanifold, and each way gives rise to a different way of defining the tangent space. The first way uses a parameterization, and the second way uses a system of equations.

Suppose that f=(f_1,...,f_n) is a local parameterization of a submanifold M in Euclidean space R^n. Say,

 f:U->R^n,
(1)

where U is the open unit ball in R^k, and f(U) subset M. At the point p=f(0), the tangent space is the image of the Jacobian of f, as a linear transformation from R^k to R^n. For example, consider the unit sphere

 S^2={(y_1,y_2,y_3):y_1^2+y_2^2+y_3^2=1}
(2)

in R^3. Then the function (with the domain U={(x_1,x_2):x_1^2+x_2^2<1})

 f=(x_1,x_2,sqrt(1-x_1^2-x_2^2))
(3)

parameterizes a neighborhood of the north pole. Its Jacobian at (0,0) is given by the matrix

 [1 0; 0 1; 0 0]
(4)

whose image is the tangent space at p,

 TS^2|_((0,0,1))={(a,b,0)}.
(5)

An alternative description of a submanifold M as the set of solutions to a system of equations leads to another description of tangent vectors. Consider a submanifold M which is the set of solutions to the system of equations

f_1(x_1,...,x_n)=0
(6)
|
(7)
f_r(x_1,...,x_n)=0,
(8)

where k+r=n and the Jacobian of f:R^n->R^r, with f=(f_1,...,f_n), has rank r at the solutions M to f=0. A tangent vector v at a solution p is an infinitesimal solution to the above equations (at p). The tangent vector v=(v_1,...,v_n) is a solution of the derivative (linearization) of f, i.e., it is in the null space of the Jacobian.

Consider this method in the recomputation the tangent space of the sphere at the north pole. The sphere is two-dimensional and is described as the solution to single equation (3-2=1) x_1^2+x_2^2+x_3^2=1. Set f_1=x_1^2+x_2^2+x_3^2-1. We want to compute the tangent space at the solution f_1(0,0,1)=0 (at the north pole). The Jacobian at this point is the 1×3 matrix [0,0,2], and its null space is the tangent space

 TS^2|_((0,0,1))={(a,b,0)}.
(9)

It appears that the tangent space depends either on the choice of parametrization, or on the choice of system of equations. Because the Jacobian of a composition of functions obeys the chain rule, the tangent space is well-defined. Note that the Jacobian of a diffeomorphism is an invertible linear map, and these correspond to the ways the equations can be changed. The basic facts from linear algebra used to show that the tangent space is well-defined are the following.

1. If A:R^k->R^k is invertible, then the image of B:R^k->R^n is the same as the image of AB.

2. If A:R^n->R^n is invertible, then the null space of B:R^n->R^r is the same as the null space of BA. More precisely, Null(BA)=A^(-1)(Null(B)).

These techniques work in any dimension. In addition, they generalize to submanifolds of an abstract manifold, because tangent vectors depend on local properties. In particular, the tangent space can be computed in any coordinate chart, because any change in coordinate chart corresponds to a diffeomorphism in Euclidean space.

The tangent space can give some geometric insight to higher-dimensional phenomena. For example, to compute the tangent space to the torus (donut) M in R^4 (which is a flat manifold), note that it can be parametrized, by

 f(x_1,x_2)=(sinx_1,cosx_1,sinx_2,cosx_2)
(10)

with domain U={(x_1,x_2):x_1^2+x_2^2<1}, near the point p=f(0,0)=(0,1,0,1). Its Jacobian at p is the matrix

 [1 0; 0 0; 0 1; 0 0],
(11)

whose image is the tangent space TM|_p={(a,0,b,0)}.

Alternatively, M is the set of solutions to equations

f_1(x_1,x_2,x_3,x_4)=x_1^2+x_2^2-1=0
(12)
f_2(x_1,x_2,x_3,x_4)=x_3^2+x_4^2-1=0.
(13)

The Jacobian at the solution p=(0,1,0,1) is the matrix

 [0 2 0 0; 0 0 0 2],
(14)

whose null space is the tangent space TM|_p={(a,0,b,0)}.


See also

Calculus, Chart Tangent Space, Coordinate Chart, Differential k-Form, Directional Derivative, Euclidean Space, Exterior Algebra, Intrinsic Tangent Space, Jacobian, Linear Algebra, Manifold, Null Space, Tangent Bundle, Tangent Plane, Tangent Vector, Vector Field, Vector Space, Velocity Vector

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Submanifold Tangent Space." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SubmanifoldTangentSpace.html

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