Tangent space
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.
of a differentiable manifold a tangent space—a real vector space that intuitively contains the possible directions in which one can tangentially pass through
This is a generalization of the notion of a vector, based at a given initial point, in a Euclidean space.
This was the traditional approach toward defining parallel transport.
Many authors in differential geometry and general relativity use it.
For example, a curve that crosses itself does not have a unique tangent line at that point.
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.
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.
The informal description above relies on a manifold's ability to be embedded into an ambient vector space
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.
In the embedded-manifold picture, a tangent vector at a point
is thought of as the velocity of a curve passing through the point
We can therefore define a tangent vector as an equivalence class of curves passing through
This defines an equivalence relation on the set of all differentiable curves initialized at
, and equivalence classes of such curves are known as tangent vectors of
turns out to be bijective and may be used to transfer the vector-space operations on
is a real associative algebra with respect to the pointwise product and sum of functions and scalar multiplication.
Generalizations of this definition are possible, for instance, to complex manifolds and algebraic varieties.
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.
manifold in a natural manner (take coordinate charts to be identity maps on open subsets of
Another way to think about tangent vectors is as directional derivatives.
, one defines the corresponding directional derivative at a point
between smooth (or differentiable) manifolds induces natural linear maps between their corresponding tangent spaces: If the tangent space is defined via differentiable curves, then this map is defined by If, instead, the tangent space is defined via derivations, then this map is defined by The linear map
It is frequently expressed using a variety of other notations: In a sense, the derivative is the best linear approximation to
coincides with the usual notion of the differential of the function
An important result regarding the derivative map is the following: Theorem — If
This is a generalization of the inverse function theorem to maps between manifolds.
