A **tangent space** is a generalization to manifolds of the simple idea of a tangent as applied to two-dimensional curves.

A manifold is a topological space that, near every point, can be modeled on Euclidean space.

**One dimensional manifold**includes lines and curves.**Two-dimensional manifolds**are surfaces: spheres and cylinders are both examples.

One way to imagine it is to think of tangent space of a point x on a manifold to be a vector space that contains every possible direction you could tangentially pass through x; a surface that contains every tangent vector of that point.

A tangent space has the same number of dimensions as its manifold does. If we put together all the tangent spaces of every point on the manifold, we get the **tangent bundle**. A tangent bundle has twice the dimension of the original manifold.

## Example of a Tangent Space.

The image above shows one manifold you are very familiar with: the sphere. It can be described by the equation

Where our point of interest is p = (0,0,0). Then the line

will be tangent to our sphere at the point p, and every vector (a,b,0) will be a tangent vector of our sphere at the point p. Remember that the collection of all tangent vectors to a manifold at a given point is the tangent space.

## References

Wijsman, Robert A. Invariant Measures on Groups and Their Use in Statistics. Chapter 3. Differentiable Manifolds, Tangent Spaces, and Vector Fields. Institute of Mathematical Statistics, 1990. Lecture Notes – Monograph Series. Retrieved from https://projecteuclid.org/download/pdf_1/euclid.lnms/1215540658 on June 25, 2019.