Meaning of tangent space | Babel Free

Definitions

An n-dimensional vector space that represents the set of all vectors tangent to given n-dimensional differentiable manifold M at point x.

Equivalents

Examples

“This gives a degenerate norm on the tangent space of Hₙ at the origin, and one can use⟳ left translations to define⟳ similar norms at the tangent spaces of Hₙ at all other points.”

“In a Lie⟳ group G, because left translation by an element g #92;in G is a diffeomorphism that maps a neighborhood of the identity to a neighborhood of g, all the local information about the group is concentrated in a neighborhood of the identity, and the tangent space of the identity assumes a special importance. Moreover, one can give⟳ the tangent space TₑG a Lie⟳ bracket [ , ], so that in addition to being a vector space, it becomes a Lie⟳ algebra, called the Lie⟳ algebra of the Lie⟳ group.”

“More dramatically, there are situations where one has a sequence of smooth manifolds such that the limit⟳ is again a smooth manifold, but with a limiting “tangent space” which has nothing to do with the actual tangent space of the limit⟳!”

CEFR level