Tangent space

The tangent space of a differentiable manifold at a point is a vector space corresponding to possible velocities when moving through . #m/def/geo/diff A number of equivalent characterizations are useful. See also Tangent map, Tangent bundle.

Intrinsic manifold

The following characterizations of are all useful.

As derivations at a point

Let , and suppose is the set of vector fields viewed as derivations. We define the tangent space as the image of the map

i.e. the set of all derivations evaluated at .

Chart-free characterization as velocities

Let , be a chart at , and

be the set of all paths such that . We define an equivalence relation on so that two paths are equivalent iff

which is easily shown to be independent of choice of .

Via the cotangent space

The cotangent space admits an intrinsic characterization, which is applicable in other (non-differentiable) settings. The tangent space is simply the dual vector space .

Equivalence of characterizations

#missing/proof

Real embedded manifold

Both the following characterizations of the tangent space of a real embedded manifold is useful.

Fixed chart characterization

Let be a Real embedded manifold and . Let

be a local parameterization at , and be its Total derivative. Then is the tangent space at .

Chart-free characterization as velocities

Let be a Real embedded manifold and . Let

be the set of all differentiable paths such that . Then the set of all “velocities at ”

is the tangent space at .

The primary advantage of the fixed chart characterization is that its vector space status is clear, whereas the chart-free characterization is more intuitive and establishes chart-independence.

Equivalence of characterizations

Take a coördinate chart with . Let and let denote the fixed-chart characterization and denote the chart-free characterization.

Let for some differentiable with . Then

Herefore .

Now let for some . Define

and let (we choose so that remains in ). It follows and . Herefore .

Thus .

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