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tangent space

From Wiktionary, the free dictionary

English

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Noun

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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 Hn at the origin, and one can use left translations to define similar norms at the tangent spaces of Hn at all other points.
    • 2011, Loring W. Tu, An Introduction to Manifolds, 2nd edition, Springer, page 178:
      In a Lie group G, because left translation by an element g 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.
    • 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!

Usage notes

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Typically denoted TxM, 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 TxM.

Holonyms

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Translations

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Further reading

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