Difference between revisions of "Tangent space"
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A linear (more rarely, affine) space of [[tangent vector|vectors tangent]] to a smooth (differentiable) [[manifold]] (curve, surface, $\dots$) $M$ at a given point $a\in M$. One of the standard notations is $T_aM$. | A linear (more rarely, affine) space of [[tangent vector|vectors tangent]] to a smooth (differentiable) [[manifold]] (curve, surface, $\dots$) $M$ at a given point $a\in M$. One of the standard notations is $T_aM$. | ||
| − | For curves, surfaces and submanifolds embedded in a Euclidean subspace $\R^n$ the tangent subspace can be identified with an affine subset (of the corresponding dimension $1,2,\dots$) in the ambient space, passing through $a$. For abstract manifolds $T_aM$ | + | For curves, surfaces and submanifolds embedded in a Euclidean subspace $\R^n$ the tangent subspace can be identified with an affine subset (of the corresponding dimension $1,2,\dots$) in the ambient space, passing through $a$. For abstract manifolds $T_aM$ is naturally isomorphic to the linear space of [[Derivation in a ring|derivations]] $D:C^\infty(M)\to\R$ of the ring of smooth functions on $M$ satisfying the [[Leibniz rule]]: |
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D\in T_a M\iff D:C^\infty(M)\to\R,\qquad D(f\pm g)=Df\pm Dg,\ D(\lambda f)=\lambda Df,\ D(f\cdot g)=f(a)\cdot Dg+g(a)\cdot Df. | D\in T_a M\iff D:C^\infty(M)\to\R,\qquad D(f\pm g)=Df\pm Dg,\ D(\lambda f)=\lambda Df,\ D(f\cdot g)=f(a)\cdot Dg+g(a)\cdot Df. | ||
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The (disjoint) union of tangent spaces to a manifold to all its points has a natural structure of a [[bundle]] over $M$, called the [[tangent bundle]]. | The (disjoint) union of tangent spaces to a manifold to all its points has a natural structure of a [[bundle]] over $M$, called the [[tangent bundle]]. | ||
| − | For submanifolds of $\R^n$ the tangent space $T_aM$ | + | For submanifolds of $\R^n$ the tangent space $T_aM$ coincides with the [[tangent cone]] defined as the union of limits of secants. |
Revision as of 06:03, 15 May 2012
A linear (more rarely, affine) space of vectors tangent to a smooth (differentiable) manifold (curve, surface, $\dots$) $M$ at a given point $a\in M$. One of the standard notations is $T_aM$.
For curves, surfaces and submanifolds embedded in a Euclidean subspace $\R^n$ the tangent subspace can be identified with an affine subset (of the corresponding dimension $1,2,\dots$) in the ambient space, passing through $a$. For abstract manifolds $T_aM$ is naturally isomorphic to the linear space of derivations $D:C^\infty(M)\to\R$ of the ring of smooth functions on $M$ satisfying the Leibniz rule: $$ D\in T_a M\iff D:C^\infty(M)\to\R,\qquad D(f\pm g)=Df\pm Dg,\ D(\lambda f)=\lambda Df,\ D(f\cdot g)=f(a)\cdot Dg+g(a)\cdot Df. $$ The (disjoint) union of tangent spaces to a manifold to all its points has a natural structure of a bundle over $M$, called the tangent bundle.
For submanifolds of $\R^n$ the tangent space $T_aM$ coincides with the tangent cone defined as the union of limits of secants.
Tangent space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent_space&oldid=26632