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''of a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t0921101.png" />''
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[[Category:TeX done]]
  
The [[Vector bundle|vector bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t0921102.png" />, also denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t0921103.png" />, whose total space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t0921104.png" /> is given by the union of the tangent spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t0921105.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t0921106.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t0921107.png" />, consisting of the tangent vectors to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t0921108.png" />, and with projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t0921109.png" /> mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211010.png" /> to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211011.png" />. A section of the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211012.png" /> is a vector field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211013.png" /> (cf. [[Vector field on a manifold|Vector field on a manifold]]). An atlas on the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211014.png" /> is defined through an atlas of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211015.png" />. The bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211016.png" /> is locally trivial. The transition functions of the tangent bundle are defined by the Jacobi matrices of the transition functions of the atlas of the manifold.
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''of a differentiable manifold $M$''
  
Associated with the tangent bundle is the frame bundle of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211017.png" />, which is a principal bundle. The bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211018.png" /> dual to the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211019.png" />, called the cotangent bundle, consists of the cotangent spaces to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211020.png" />. Its sections are the differential or Pfaffian forms (cf. [[Pfaffian form|Pfaffian form]]).
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The [[Vector bundle|vector bundle]] $\tau:TM\rightarrow M$, also denoted $\tau (M)$, whose total space $TM$ is given by the union of the tangent spaces $TM|_x$ to $M$ at $x\in M$, consisting of the tangent vectors to $M$, and with projection $\tau$ mapping $TM|_x$ to the point $x$. A section of the tangent bundle $\tau (M)$ is a vector field on $M$ (cf. [[Vector field on a manifold|Vector field on a manifold]]). An atlas on the manifold $TM$ is defined through an atlas of the manifold $M$. The bundle $\tau (M)$ is locally trivial. The transition functions of the tangent bundle are defined by the Jacobi matrices of the transition functions of the atlas of the manifold.
  
A differentiable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211021.png" /> induces a morphism of tangent bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211022.png" />; the corresponding mapping of the total spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211023.png" /> is called the tangent mapping to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211024.png" /> (or differential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211025.png" />). In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211026.png" /> is an immersion (cf. [[Immersion of a manifold|Immersion of a manifold]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211027.png" /> is a subbundle of the induced vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211028.png" />. The quotient bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211029.png" /> is called the normal bundle of the immersion. Dually, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211030.png" /> is a [[Submersion|submersion]], then the quotient bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211031.png" /> is called a subbundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211032.png" />. If one chooses for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211036.png" /> respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211038.png" /> is called the tangent bundle of second order.
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Associated with the tangent bundle is the frame bundle of the manifold $M$, which is a principal bundle. The bundle $\tau^*(M)$ dual to the tangent bundle $\tau (M)$, called the cotangent bundle, consists of the cotangent spaces to $M$. Its sections are the differential or Pfaffian forms (cf. [[Pfaffian form|Pfaffian form]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211039.png" /> is trivial, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211040.png" /> is called a parallelizable manifold.
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A differentiable mapping $h:M\rightarrow N$ induces a morphism of tangent bundles $\tau(M)\rightarrow \tau(N)$; the corresponding mapping of the total spaces $Th:TM\rightarrow TN$ is called the tangent mapping to $h$ (or differential of $h$). In particular, if $i:M\rightarrow N$ is an immersion (cf. [[Immersion of a manifold|Immersion of a manifold]]), then $\tau(M)$ is a subbundle of the induced vector bundle $i^*\tau(N)$. The quotient bundle $\nu (i)=i^*\tau(N)/\tau(M)$ is called the normal bundle of the immersion. Dually, if $j:M\rightarrow N$ is a [[Submersion|submersion]], then the quotient bundle $\tau(M)/j^*\tau(N)$ is called a subbundle of $j$. If one chooses for $M$ and $N$, $TM$ and $M$ respectively, and $h=\tau:TM\rightarrow M$, then $\tau^*\tau (M)$ is called the tangent bundle of second order.
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If $\tau (M)$ is trivial, then $M$ is called a parallelizable manifold.
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
The tangent mapping (also called differential) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211041.png" /> induced by a differentiable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211042.png" /> is given by
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The tangent mapping (also called differential) $T\alpha (m):T_mM\rightarrow T_{\alpha(m)}N$ induced by a differentiable mapping $\alpha$ is given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211043.png" /></td> </tr></table>
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$$T\alpha(m)(v)(g)=v(g\alpha),$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211047.png" /> is the algebra of smooth functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211048.png" /> and a [[Tangent vector|tangent vector]] is seen as a special kind of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211049.png" />-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211050.png" />.
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$g:N\rightarrow \mathbb{R}$, $\alpha:M\rightarrow N$, $v:F(M)\rightarrow \mathbb{R}$, where $F(M)$ is the algebra of smooth functions on $M$ and a [[Tangent vector|tangent vector]] is seen as a special kind of $\mathbb{R}$-linear mapping $F(M)\rightarrow\mathbb{R}$.
  
In terms of local coordinates and the  " /xi-notation"  (cf. [[Tangent vector|Tangent vector]]), the matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211051.png" /> is given by the Jacobian matrix of the expression for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211052.png" /> in the local coordinates.
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In terms of local coordinates and the  " /xi-notation"  (cf. [[Tangent vector|Tangent vector]]), the matrix of $T\alpha (m)$ is given by the Jacobian matrix of the expression for $052$ in the local coordinates.
  
There are many notations in use for the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211053.png" />. Some common ones are: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211058.png" />. The last one, in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211059.png" /> is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211060.png" />,  "sort of agrees"  in both notation and name with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211061.png" /> as the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211062.png" />-form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211063.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211064.png" /> (cf. [[Differential|Differential]]; [[Differential form|Differential form]]). Using the  " /xi and dxi"  notation (cf. [[Tangent vector|Tangent vector]]), the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211065.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211066.png" /> is given in local coordinates by
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There are many notations in use for the differential $T\alpha :TM\rightarrow TN$. Some common ones are: $T\alpha$, $\alpha*$, $J(\alpha)$, $D\alpha$, $d\alpha$. The last one, in case $\alpha$ is a function $\alpha:M\rightarrow \mathbb{R}$,  "sort of agrees"  in both notation and name with $d\alpha$ as the differential $1$-form on $M$ defined by $\alpha$ (cf. [[Differential|Differential]]; [[Differential form|Differential form]]). Using the  " /xi and dxi"  notation (cf. [[Tangent vector|Tangent vector]]), the differential $1$-form $d\alpha$ is given in local coordinates by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211067.png" /></td> </tr></table>
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$$d\alpha=\frac{\partial\alpha}{\partial x_1}dx_1+\dots+\frac{\partial\alpha}{\partial x_n}dx_n$$
  
(where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211068.png" /> is the result of applying the tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211069.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211070.png" />). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211071.png" /> be the coordinate for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211072.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211073.png" /> is given by
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(where $\partial\alpha/\partial x_i$ is the result of applying the tangent vector $\partial/\partial x_i$ to $\alpha$). Let $t$ be the coordinate for $\mathbb{R}$. Then $d\alpha:T_mM\rightarrow T_{\alpha(m)}\mathbb{R}$ is given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211074.png" /></td> </tr></table>
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$$d\alpha(\frac{\partial}{\partial x_i})=\left(\frac{\partial\alpha}{\partial x_i}\right)\frac{\partial}{\partial t}=$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211075.png" /></td> </tr></table>
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$$=\left(\frac{\partial\alpha}{\partial x_1}dx_1+\dots+\frac{\partial\alpha}{\partial x_n}dx_n\right)(\frac{\partial}{\partial x_i})\frac{\partial}{\partial t},$$
  
because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211076.png" />.
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because $dx_j(\partial/\partial x_k)=\delta_{jk}$.
  
The differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211078.png" />-frame bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211079.png" /> has as fibre over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211080.png" /> the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211081.png" />-frames in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211082.png" />. (An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211084.png" />-frame at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211085.png" /> is a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211086.png" /> independent vectors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211087.png" />. The frame bundle is the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211088.png" />-frame bundle. A frame on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092110/t09211089.png" /> is a section of the frame bundle and a framed manifold is a manifold provided with a frame.)
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The differential $r$-frame bundle over $M$ has as fibre over $m$ the set of all $r$-frames in $T_mM$. (An $r$-frame at $m\in M$ is a set of $r$ independent vectors in $T_mM$. The frame bundle is the differential $n$-frame bundle. A frame on $M$ is a section of the frame bundle and a framed manifold is a manifold provided with a frame.)
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,  "Calculus on manifolds" , Benjamin/Cummings  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.W. Hirsch,  "Differential topology" , Springer  (1976)  pp. Chapt. 5, Sect. 3</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Brickell,  R.S. Clark,  "Differentiable manifolds" , v. Nostrand-Reinhold  (1970)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Auslander,  R.E. MacKenzie,  "Introduction to differentiable manifolds" , Dover, reprint  (1977)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Hermann,  "Geometry, physics, and systems" , M. Dekker  (1973)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  Yu. Borisovich,  N. Bliznyakov,  Ya. Izrailevich,  T. Fomenko,  "Introduction to topology" , Kluwer  (1993)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,  "Calculus on manifolds" , Benjamin/Cummings  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.W. Hirsch,  "Differential topology" , Springer  (1976)  pp. Chapt. 5, Sect. 3</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Brickell,  R.S. Clark,  "Differentiable manifolds" , v. Nostrand-Reinhold  (1970)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Auslander,  R.E. MacKenzie,  "Introduction to differentiable manifolds" , Dover, reprint  (1977)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Hermann,  "Geometry, physics, and systems" , M. Dekker  (1973)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  Yu. Borisovich,  N. Bliznyakov,  Ya. Izrailevich,  T. Fomenko,  "Introduction to topology" , Kluwer  (1993)  (Translated from Russian)</TD></TR></table>

Revision as of 07:44, 20 May 2017


of a differentiable manifold $M$

The vector bundle $\tau:TM\rightarrow M$, also denoted $\tau (M)$, whose total space $TM$ is given by the union of the tangent spaces $TM|_x$ to $M$ at $x\in M$, consisting of the tangent vectors to $M$, and with projection $\tau$ mapping $TM|_x$ to the point $x$. A section of the tangent bundle $\tau (M)$ is a vector field on $M$ (cf. Vector field on a manifold). An atlas on the manifold $TM$ is defined through an atlas of the manifold $M$. The bundle $\tau (M)$ is locally trivial. The transition functions of the tangent bundle are defined by the Jacobi matrices of the transition functions of the atlas of the manifold.

Associated with the tangent bundle is the frame bundle of the manifold $M$, which is a principal bundle. The bundle $\tau^*(M)$ dual to the tangent bundle $\tau (M)$, called the cotangent bundle, consists of the cotangent spaces to $M$. Its sections are the differential or Pfaffian forms (cf. Pfaffian form).

A differentiable mapping $h:M\rightarrow N$ induces a morphism of tangent bundles $\tau(M)\rightarrow \tau(N)$; the corresponding mapping of the total spaces $Th:TM\rightarrow TN$ is called the tangent mapping to $h$ (or differential of $h$). In particular, if $i:M\rightarrow N$ is an immersion (cf. Immersion of a manifold), then $\tau(M)$ is a subbundle of the induced vector bundle $i^*\tau(N)$. The quotient bundle $\nu (i)=i^*\tau(N)/\tau(M)$ is called the normal bundle of the immersion. Dually, if $j:M\rightarrow N$ is a submersion, then the quotient bundle $\tau(M)/j^*\tau(N)$ is called a subbundle of $j$. If one chooses for $M$ and $N$, $TM$ and $M$ respectively, and $h=\tau:TM\rightarrow M$, then $\tau^*\tau (M)$ is called the tangent bundle of second order.

If $\tau (M)$ is trivial, then $M$ is called a parallelizable manifold.

References

[1] C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969)


Comments

The tangent mapping (also called differential) $T\alpha (m):T_mM\rightarrow T_{\alpha(m)}N$ induced by a differentiable mapping $\alpha$ is given by

$$T\alpha(m)(v)(g)=v(g\alpha),$$

$g:N\rightarrow \mathbb{R}$, $\alpha:M\rightarrow N$, $v:F(M)\rightarrow \mathbb{R}$, where $F(M)$ is the algebra of smooth functions on $M$ and a tangent vector is seen as a special kind of $\mathbb{R}$-linear mapping $F(M)\rightarrow\mathbb{R}$.

In terms of local coordinates and the " /xi-notation" (cf. Tangent vector), the matrix of $T\alpha (m)$ is given by the Jacobian matrix of the expression for $052$ in the local coordinates.

There are many notations in use for the differential $T\alpha :TM\rightarrow TN$. Some common ones are: $T\alpha$, $\alpha*$, $J(\alpha)$, $D\alpha$, $d\alpha$. The last one, in case $\alpha$ is a function $\alpha:M\rightarrow \mathbb{R}$, "sort of agrees" in both notation and name with $d\alpha$ as the differential $1$-form on $M$ defined by $\alpha$ (cf. Differential; Differential form). Using the " /xi and dxi" notation (cf. Tangent vector), the differential $1$-form $d\alpha$ is given in local coordinates by

$$d\alpha=\frac{\partial\alpha}{\partial x_1}dx_1+\dots+\frac{\partial\alpha}{\partial x_n}dx_n$$

(where $\partial\alpha/\partial x_i$ is the result of applying the tangent vector $\partial/\partial x_i$ to $\alpha$). Let $t$ be the coordinate for $\mathbb{R}$. Then $d\alpha:T_mM\rightarrow T_{\alpha(m)}\mathbb{R}$ is given by

$$d\alpha(\frac{\partial}{\partial x_i})=\left(\frac{\partial\alpha}{\partial x_i}\right)\frac{\partial}{\partial t}=$$

$$=\left(\frac{\partial\alpha}{\partial x_1}dx_1+\dots+\frac{\partial\alpha}{\partial x_n}dx_n\right)(\frac{\partial}{\partial x_i})\frac{\partial}{\partial t},$$

because $dx_j(\partial/\partial x_k)=\delta_{jk}$.

The differential $r$-frame bundle over $M$ has as fibre over $m$ the set of all $r$-frames in $T_mM$. (An $r$-frame at $m\in M$ is a set of $r$ independent vectors in $T_mM$. The frame bundle is the differential $n$-frame bundle. A frame on $M$ is a section of the frame bundle and a framed manifold is a manifold provided with a frame.)

References

[a1] M. Spivak, "Calculus on manifolds" , Benjamin/Cummings (1965)
[a2] M.W. Hirsch, "Differential topology" , Springer (1976) pp. Chapt. 5, Sect. 3
[a3] F. Brickell, R.S. Clark, "Differentiable manifolds" , v. Nostrand-Reinhold (1970)
[a4] L. Auslander, R.E. MacKenzie, "Introduction to differentiable manifolds" , Dover, reprint (1977)
[a5] R. Hermann, "Geometry, physics, and systems" , M. Dekker (1973)
[a6] Yu. Borisovich, N. Bliznyakov, Ya. Izrailevich, T. Fomenko, "Introduction to topology" , Kluwer (1993) (Translated from Russian)
How to Cite This Entry:
Tangent bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent_bundle&oldid=41514
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article