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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 $T_aM$ can be identified with 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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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]]:
 
$$
 
$$
 
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 $T_aM$ can be alternatively defined as the union of limit points of secants passing through the point $a\in M$: for smooth submanifolds this gives the same definition as before, but if $a$ is a singular point on $M$, the resulting [[tangent cone]] may be a non-affine subset.
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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.

How to Cite This Entry:
Tangent space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent_space&oldid=26632