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''of a submanifold''
 
''of a submanifold''
  
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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The vector bundle consisting of tangent vectors to the ambient manifold that are normal to the submanifold. If $  X $
 +
is a [[Riemannian manifold|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|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 }  $,  
 +
$  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 <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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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:
  
<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>
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$$
 +
N _ {Y/X}  = \
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f ^ { * } T _ {X} / T _ {Y} .
 +
$$
  
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" />.
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Similarly one defines the normal bundle $  N _ {Y/X} $
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of a non-singular algebraic subvariety $  Y $
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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 $;  
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it is an algebraic (or analytic) vector bundle over $  Y $
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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 $
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that determines the divisor $  Y $.
  
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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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">[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></table>
 
<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></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><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>
 
<table><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>

Revision as of 08:03, 6 June 2020


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

Comments

References

[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=23914
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article