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''of a submanifold''
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The vector bundle consisting of tangent vectors to the ambient manifold that are normal to the submanifold. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n0674101.png" /> is a [[Riemannian manifold|Riemannian manifold]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n0674102.png" /> is an (immersed) submanifold of it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n0674103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n0674104.png" /> are the tangent bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n0674105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n0674106.png" /> (cf. [[Tangent bundle|Tangent bundle]]), then the normal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n0674107.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n0674108.png" /> is the subbundle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n0674109.png" /> consisting of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741011.png" />, that are orthogonal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741012.png" />.
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With the help of normal bundles one constructs, for example, tubular neighbourhoods of submanifolds (cf. [[Tubular neighbourhood|Tubular neighbourhood]]). The normal bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741013.png" />, regarded up to equivalence, does not depend on the choice of the Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741014.png" />, since it can be defined without recourse to the metric as the quotient bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741015.png" /> of the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741016.png" /> restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741017.png" /> by the vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741018.png" />. Somewhat more general is the construction of the normal bundle of an arbitrary immersion (cf. [[Immersion of a manifold|Immersion of a manifold]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741019.png" /> of differentiable manifolds:
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''of a submanifold''
 
 
<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/n/n067/n067410/n06741020.png" /></td> </tr></table>
 
 
 
Similarly one defines the normal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741021.png" /> of a non-singular algebraic subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741022.png" /> in a non-singular [[Algebraic variety|algebraic variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741023.png" /> or that of an analytic submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741024.png" /> in an [[Analytic manifold|analytic manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741025.png" />; it is an algebraic (or analytic) vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741026.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741027.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741028.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741029.png" /> is isomorphic to the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741030.png" /> of the bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741031.png" /> that determines the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741032.png" />.
 
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741033.png" /> is an analytic subspace of an analytic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741034.png" />, the normal bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741035.png" /> is sometimes defined as the analytic family of vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741036.png" /> dual to the conormal sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067410/n06741037.png" /> (see [[Normal sheaf|Normal sheaf]]). For applications of normal bundles to the problem of contractibility of submanifolds see [[Exceptional analytic set|Exceptional analytic set]]; [[Exceptional subvariety|Exceptional subvariety]].
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The vector bundle consisting of tangent vectors to the ambient manifold that are normal to the submanifold. If  $  X $
 
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is a [[Riemannian manifold|Riemannian manifold]],  $  Y $
====References====
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is an (immersed) submanifold of it,  $  T _ {X} $
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.L. Onishchik,   "Pseudoconvexity in the theory of complex spaces" ''J. Soviet Math.'' , '''14''' : 3  (1980) pp. 1363–1406 ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''15''' (1977) pp. 93–156</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.W. Milnor,   J.D. Stasheff,   "Characteristic classes" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.A. Rokhlin,  D.B. Fuks,  "Beginner's course in topology. Geometric chapters" , Springer (1984) (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.W. Hirsch,   "Differential topology" , Springer  (1976)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.R. Shafarevich,  "Basic algebraic geometry" , Springer  (1977)  (Translated from Russian)</TD></TR></table>
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and  $  T _ {Y} $
 +
are the tangent bundles over  $  X $
 +
and  $  Y $(
 +
cf. [[Tangent bundle|Tangent bundle]]), then the normal bundle  $  N _ {Y/X} $
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of  $ Y $
 +
is the subbundle in $ T _ {X} \mid _ {Y} $
 +
consisting of the vectors $ u \in T _ {X,y }  $,  
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$ y \in Y $,  
 +
that are orthogonal to $ T _ {Y,y} $.
  
 +
With the help of normal bundles one constructs, for example, tubular neighbourhoods of submanifolds (cf. [[Tubular neighbourhood|Tubular neighbourhood]]). The normal bundle over  $  Y $,
 +
regarded up to equivalence, does not depend on the choice of the Riemannian metric on  $  X $,
 +
since it can be defined without recourse to the metric as the quotient bundle  $  T _ {X} \mid  _ {Y} / T _ {Y} $
 +
of the tangent bundle  $  T _ {X} $
 +
restricted to  $  Y $
 +
by the vector bundle  $  T _ {Y} $.
 +
Somewhat more general is the construction of the normal bundle of an arbitrary immersion (cf. [[Immersion of a manifold|Immersion of a manifold]])  $  f:  Y \rightarrow X $
 +
of differentiable manifolds:
  
 +
$$
 +
N _ {Y/X}  = \
 +
f ^ { * } T _ {X} / T _ {Y} .
 +
$$
  
====Comments====
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Similarly one defines the normal bundle  $  N _ {Y/X} $
 +
of a non-singular algebraic subvariety  $  Y $
 +
in a non-singular [[Algebraic variety|algebraic variety]]  $  \overline{X}\; $
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or that of an analytic submanifold  $  Y $
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in an [[Analytic manifold|analytic manifold]]  $  X $;
 +
it is an algebraic (or analytic) vector bundle over  $  Y $
 +
of rank  $  \mathop{\rm codim}  Y $.
 +
In particular, if  $  \mathop{\rm codim}  Y = 1 $,
 +
then  $  N _ {Y/X} $
 +
is isomorphic to the restriction to  $  Y $
 +
of the bundle over  $  X $
 +
that determines the divisor  $  Y $.
  
 +
When  $  Y $
 +
is an analytic subspace of an analytic space  $  ( X, {\mathcal O} _ {X} ) $,
 +
the normal bundle of  $  Y $
 +
is sometimes defined as the analytic family of vector spaces  $  N _ {Y/X} \rightarrow Y $
 +
dual to the conormal sheaf  $  N _ {Y/X}  ^ {*} $(
 +
see [[Normal sheaf|Normal sheaf]]). For applications of normal bundles to the problem of contractibility of submanifolds see [[Exceptional analytic set|Exceptional analytic set]]; [[Exceptional subvariety|Exceptional subvariety]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.E. Steenrod,   "The topology of fibre bundles" , Princeton Univ. Press (1951)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.L. Onishchik, "Pseudoconvexity in the theory of complex spaces" ''J. Soviet Math.'' , '''14''' : 3 (1980) pp. 1363–1406 ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''15''' (1977) pp. 93–156 {{MR|}} {{ZBL|0449.32020}} </TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974) {{MR|0440554}} {{ZBL|0298.57008}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.A. Rokhlin, D.B. Fuks, "Beginner's course in topology. Geometric chapters" , Springer (1984) (Translated from Russian) {{MR|759162}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M.W. Hirsch, "Differential topology" , Springer (1976) {{MR|0448362}} {{ZBL|0356.57001}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951) {{MR|0039258}} {{ZBL|0054.07103}} </TD></TR></table>

Latest revision as of 06:02, 18 April 2023


of a submanifold

The vector bundle consisting of tangent vectors to the ambient manifold that are normal to the submanifold. If $ X $ is a Riemannian manifold, $ Y $ is an (immersed) submanifold of it, $ T _ {X} $ and $ T _ {Y} $ are the tangent bundles over $ X $ and $ Y $( cf. Tangent bundle), then the normal bundle $ N _ {Y/X} $ of $ Y $ is the subbundle in $ T _ {X} \mid _ {Y} $ consisting of the vectors $ u \in T _ {X,y } $, $ y \in Y $, that are orthogonal to $ T _ {Y,y} $.

With the help of normal bundles one constructs, for example, tubular neighbourhoods of submanifolds (cf. Tubular neighbourhood). The normal bundle over $ Y $, regarded up to equivalence, does not depend on the choice of the Riemannian metric on $ X $, since it can be defined without recourse to the metric as the quotient bundle $ T _ {X} \mid _ {Y} / T _ {Y} $ of the tangent bundle $ T _ {X} $ restricted to $ Y $ by the vector bundle $ T _ {Y} $. Somewhat more general is the construction of the normal bundle of an arbitrary immersion (cf. Immersion of a manifold) $ f: Y \rightarrow X $ of differentiable manifolds:

$$ N _ {Y/X} = \ f ^ { * } T _ {X} / T _ {Y} . $$

Similarly one defines the normal bundle $ N _ {Y/X} $ of a non-singular algebraic subvariety $ Y $ in a non-singular algebraic variety $ \overline{X}\; $ or that of an analytic submanifold $ Y $ in an analytic manifold $ X $; it is an algebraic (or analytic) vector bundle over $ Y $ of rank $ \mathop{\rm codim} Y $. In particular, if $ \mathop{\rm codim} Y = 1 $, then $ N _ {Y/X} $ is isomorphic to the restriction to $ Y $ of the bundle over $ X $ that determines the divisor $ Y $.

When $ Y $ is an analytic subspace of an analytic space $ ( X, {\mathcal O} _ {X} ) $, the normal bundle of $ Y $ is sometimes defined as the analytic family of vector spaces $ N _ {Y/X} \rightarrow Y $ dual to the conormal sheaf $ N _ {Y/X} ^ {*} $( see Normal sheaf). For applications of normal bundles to the problem of contractibility of submanifolds see Exceptional analytic set; Exceptional subvariety.

References

[1] A.L. Onishchik, "Pseudoconvexity in the theory of complex spaces" J. Soviet Math. , 14 : 3 (1980) pp. 1363–1406 Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 15 (1977) pp. 93–156 Zbl 0449.32020
[2] J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974) MR0440554 Zbl 0298.57008
[3] V.A. Rokhlin, D.B. Fuks, "Beginner's course in topology. Geometric chapters" , Springer (1984) (Translated from Russian) MR759162
[4] M.W. Hirsch, "Differential topology" , Springer (1976) MR0448362 Zbl 0356.57001
[5] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[a1] N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951) MR0039258 Zbl 0054.07103
How to Cite This Entry:
Normal bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_bundle&oldid=14667
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article