tangentspace

Tangentspace is a term used in differential geometry to denote the tangent space at a point on a differentiable manifold. At a point p on a smooth n-dimensional manifold M, the tangentspace T_p M is a real vector space that encodes the possible directions in which one can move from p along smooth curves on M. It serves as a linear model of the manifold’s local structure near p and is fundamental for defining directions, velocities, and first-order changes.

There are two common constructions of T_p M. One defines tangent vectors as equivalence classes of smooth curves through p, where two curves are equivalent if they have the same velocity at p. The other defines tangent vectors as derivations: linear maps from the algebra of smooth functions on M to the real numbers that satisfy the Leibniz rule. In local coordinates, a basis for T_p M is given by the partial derivative operators ∂/∂x^i evaluated at p, and the dimension of T_p M equals the dimension of the manifold.

For an embedded submanifold of Euclidean space, T_p M can be realized concretely as the subspace of R^m consisting of velocity vectors of curves on M that pass through p. The collection of all tangent spaces forms the tangent bundle, a smooth vector bundle over M.

Applications of tangentspace include the definition of differential maps and pushforwards, analysis of curves and geodesics, and methods in robotics, physics, and optimization on manifolds. The concept is often referred to simply as the tangent space in many texts, with tangentspace used as an alternative spelling or synonym.