Tangent space: Difference between revisions
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The '''tangent space''' of a [[manifold_(geometry)|differentiable manifold]] M is a [[vector space]] at a point p on the manifold whose elements are the tangent vectors (or velocities) to the curves passing through that point p. The tangent space at this point p is usually denoted <math>T_pM</math>. | The '''tangent space''' of a [[manifold_(geometry)|differentiable manifold]] M is a [[vector space]] at a point p on the manifold whose elements are the tangent vectors (or velocities) to the curves passing through that point p. The tangent space at this point p is usually denoted <math>T_pM</math>. | ||
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:<math>\frac{\partial}{\partial x^i} (f \circ \psi^{-1})(\psi(p)).</math> | :<math>\frac{\partial}{\partial x^i} (f \circ \psi^{-1})(\psi(p)).</math> | ||
For an arbitrary curve <math>\scriptstyle \gamma, \, \psi \circ \gamma \, = \, (\gamma_1, \, \cdots, \, \gamma_n), </math> then | |||
:<math> \gamma'(0)(f) = \sum_{i=1}^{n} \frac{\partial}{\partial x^i} (f \circ \psi^{-1})(\psi(p)) \cdot \gamma_i'(0) </math> | |||
which is simply | |||
:<math> \sum^{n}_{i=1} \gamma'_{i}(0) \cdot \frac{\partial}{\partial x^i}\bigg|_{p} (f)</math> | |||
so | |||
:<math> \gamma'(0) = \sum^{n}_{i=1} \gamma'_{i}(0) \cdot \frac{\partial}{\partial x^i}\bigg|_{p} </math> | |||
as f is arbitrary. |
Revision as of 00:52, 15 November 2007
The tangent space of a differentiable manifold M is a vector space at a point p on the manifold whose elements are the tangent vectors (or velocities) to the curves passing through that point p. The tangent space at this point p is usually denoted .
The tangent space is necessary for a manifold because it offers a way in which tangent vectors at different points on the manifold can be compared (via an affine connection). If the manifold is a submanifold of , then the tangent space at a point can be thought of as an n-dimensional hyperplane at that point. However, this ambient Euclidean space is unnecessary to the definition of the tangent space.
The tangent space at a point has the same dimension as the manifold, and the union of all the tangent spaces of a manifold is called the tangent bundle, which itself is a manifold of twice the dimension of M.
Definition
Although it is tempting to define a tangent space as a "space where tangent vectors live", without a definition of a tangent space there is no definition of a tangent vector. There are various ways in which a tangent space can be defined, the most intuitive of which is in terms of directional derivatives; the space is the space identified with directional derivatives at p.
Directional derivative
A curve on the manifold is defined as a differentiable map . Let . If one defines to be all the functions that are differentiable at the point p, then one can interpret
to be an operator such that
and is a directional derivative of f in the direction of the curve . This operator can be interpreted as a tangent vector. The tangent space is then the set of all directional derivatives of curves at the point p.
Directional derivatives as a vector space
If this definition is reasonable, then the directional derivatives, must form a vector space of the same dimension as the n-dimensional manifold M. The easiest way to show this is to show that the directional derivatives form a basis of the vector space, and in order to do so, one needs to introduce a coordinate chart (see differentiable manifold).
Let where , be a coordinate chart, and . The most obvious candidates for basis vectors would be the directional derivatives along the coordinate curves, i.e. the i-th coordinate curve would be
where , the 1 being in the i-th position.
The directional derivative along a coordinate curve can be represented as
because
which becomes, via the chain rule,
For an arbitrary curve then
which is simply
so
as f is arbitrary.