tangent space

tangent space (plural tangent spaces)

1. (topology, differential topology) An n-dimensional vector space that represents the set of all vectors tangent to given n-dimensional differentiable manifold M at point x.
   • 2001, Stephen Semmes, Some Novel Types of Fractal Geometry, Oxford University Press, page 124,

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

   • 2011, Loring W. Tu, An Introduction to Manifolds, Springer, 2nd Edition, page 178,

     In a Lie group G, because left translation by an element g ${\displaystyle \in }$ G is a diffeomorphism that maps a neghborhood 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 T_eG 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.

   • 2014, Peter K. Friz, Martin Hairer, A Course on Rough Paths: With an Introduction to Regularity Structures, Springer, page 7,

     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!

Typically denoted T_xM, although the manifold may be omitted if understood.

More strictly, for any smooth curve in M passing through x, its derivative at x is a vector in T_xM.

• tangent bundle

• cotangent space

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• Differential geometry on Wikipedia.Wikipedia