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An [[Analytic space|analytic space]] over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r0800301.png" /> of real numbers. Unlike in the case of complex-analytic spaces, the structure sheaves of real-analytic spaces need not be coherent (cf. [[Coherent sheaf|Coherent sheaf]]). Real-analytic spaces are said to be coherent if their structure sheaves are coherent. All real-analytic manifolds (i.e. smooth real-analytic spaces) are coherent real-analytic spaces.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r0800302.png" /> be the germ at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r0800303.png" /> of a real-analytic subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r0800304.png" /> (cf. [[Analytic set|Analytic set]]). This defines the germ at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r0800305.png" /> of a complex-analytic subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r0800306.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r0800307.png" /> with the following equivalent properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r0800308.png" /> is the intersection of all germs of complex-analytic sets containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r0800309.png" />; 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003010.png" /> is the analytic algebra of the germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003012.png" /> is the analytic algebra of the germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003013.png" />. The germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003014.png" /> is said to be the complexification of the germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003015.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003016.png" /> is said to be the real part of the germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003017.png" />. Analogously, for any coherent real-analytic countably-infinite space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003018.png" /> it is possible to construct the complexification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003019.png" />, which is a complex-analytic space. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003020.png" /> will then have a fundamental system of neighbourhoods in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003021.png" /> which are Stein spaces (cf. [[Stein space|Stein space]]).
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The theory of coherent real-analytic spaces is similar to the theory of complex Stein spaces. Global sections of any [[Coherent analytic sheaf|coherent analytic sheaf]] of modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003022.png" /> on a coherent real-analytic countably-infinite space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003023.png" /> generate modules of germs of its sections at any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003024.png" />, and all groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003025.png" /> vanish if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003026.png" />.
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An [[Analytic space|analytic space]] over the field  $  \mathbf R $
 +
of real numbers. Unlike in the case of complex-analytic spaces, the structure sheaves of real-analytic spaces need not be coherent (cf. [[Coherent sheaf|Coherent sheaf]]). Real-analytic spaces are said to be coherent if their structure sheaves are coherent. All real-analytic manifolds (i.e. smooth real-analytic spaces) are coherent real-analytic spaces.
  
For any finite-dimensional coherent real-analytic countably-infinite space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003027.png" /> there exists a morphism
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Let  $  V _ {a} $
 +
be the germ at a point  $  a $
 +
of a real-analytic subset of  $  \mathbf R  ^ {n} $(
 +
cf. [[Analytic set|Analytic set]]). This defines the germ at  $  a $
 +
of a complex-analytic subset  $  {\widetilde{V}  } _ {a} $
 +
of the space  $  \mathbf C  ^ {n} $
 +
with the following equivalent properties: 1)  $  \widetilde{V}  _ {a} $
 +
is the intersection of all germs of complex-analytic sets containing  $  V _ {a} $;
 +
2) if  $  {\mathcal O} _ {V _ {a}  } $
 +
is the analytic algebra of the germ  $  V _ {a} $,
 +
then  $  {\mathcal O} _ {V _ {a}  } \otimes \mathbf C $
 +
is the analytic algebra of the germ  $  \widetilde{V}  _ {a} $.
 +
The germ  $  \widetilde{V}  _ {a} $
 +
is said to be the complexification of the germ  $  V _ {a} $,
 +
and  $  V _ {a} $
 +
is said to be the real part of the germ  $  \widetilde{V}  _ {a} $.
 +
Analogously, for any coherent real-analytic countably-infinite space $  X $
 +
it is possible to construct the complexification  $  \widetilde{X}  $,
 +
which is a complex-analytic space. $  X $
 +
will then have a fundamental system of neighbourhoods in  $  \widetilde{X}  $
 +
which are Stein spaces (cf. [[Stein space|Stein space]]).
  
<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/r/r080/r080030/r08003028.png" /></td> </tr></table>
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The theory of coherent real-analytic spaces is similar to the theory of complex Stein spaces. Global sections of any [[Coherent analytic sheaf|coherent analytic sheaf]] of modules  $  F $
 +
on a coherent real-analytic countably-infinite space  $  X $
 +
generate modules of germs of its sections at any point of  $  X $,
 +
and all groups  $  H  ^ {q} ( X, F  ) $
 +
vanish if  $  q \geq  1 $.
  
such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003029.png" /> is a proper one-to-one mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003030.png" /> into a coherent subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003031.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003032.png" /> is an imbedding at the smooth points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003033.png" />. In particular, any (Hausdorff and countably-infinite) real-analytic manifold is isomorphic to a real-analytic submanifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003034.png" />. For a reduced coherent real-analytic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003035.png" /> the set of classes of isomorphic real-analytic principal fibre bundles with a real structure Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003036.png" />, admitting complexification, and base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003037.png" /> is in one-to-one correspondence with the set of classes of isomorphic topological principal fibre bundles with the same structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080030/r08003038.png" />.
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For any finite-dimensional coherent real-analytic countably-infinite space  $  ( X, {\mathcal O} _ {X} ) $
 +
there exists a morphism
 +
 
 +
$$
 +
f = ( f _ {0} , f _ {1} ):
 +
( X, {\mathcal O} _ {X} )  \rightarrow \
 +
( \mathbf R  ^ {n} , {\mathcal O} _ {\mathbf R  ^ {n}  } )
 +
$$
 +
 
 +
such that $  f _ {0} $
 +
is a proper one-to-one mapping of $  X $
 +
into a coherent subspace in $  \mathbf R  ^ {n} $,  
 +
while $  f $
 +
is an imbedding at the smooth points of $  X $.  
 +
In particular, any (Hausdorff and countably-infinite) real-analytic manifold is isomorphic to a real-analytic submanifold in $  \mathbf R  ^ {n} $.  
 +
For a reduced coherent real-analytic space $  X $
 +
the set of classes of isomorphic real-analytic principal fibre bundles with a real structure Lie group $  G $,  
 +
admitting complexification, and base $  X $
 +
is in one-to-one correspondence with the set of classes of isomorphic topological principal fibre bundles with the same structure group $  G $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Tognoli,  "Some results in the theory of real analytic spaces"  M. Jurchesan (ed.) , ''Espaces Analytiques (Bucharest 1969)'' , Acad. Roumanie  (1971)  pp. 149–157</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Tognoli,  "Some results in the theory of real analytic spaces"  M. Jurchesan (ed.) , ''Espaces Analytiques (Bucharest 1969)'' , Acad. Roumanie  (1971)  pp. 149–157</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Cartan,  "Variétés analytiques réelles et variétés analytiques complexes"  ''Bull. Soc. Math. France'' , '''85'''  (1957)  pp. 77–99</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Bruhat,  H. Cartan,  "Sur la structure des sous-ensembles analytiques réels"  ''C.R. Acad. Sci. Paris'' , '''244'''  (1957)  pp. 988–900</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Bruhat,  H. Cartan,  "Sur les composantes irréductibles d'un sous-ensemble"  ''C.R. Acad. Sci. Paris'' , '''244'''  (1957)  pp. 1123–1126</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F. Bruhat,  H. Whitney,  "Quelques propriétés fondamentales des ensembles analytiques-réels"  ''Comm. Math. Helv.'' , '''33'''  (1959)  pp. 132–160</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Narasimhan,  "Introduction to the theory of analytic spaces" , ''Lect. notes in math.'' , '''25''' , Springer  (1966)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Grauert,  R. Remmert,  "Theory of Stein spaces" , Springer  (1979)  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Cartan,  "Variétés analytiques réelles et variétés analytiques complexes"  ''Bull. Soc. Math. France'' , '''85'''  (1957)  pp. 77–99</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Bruhat,  H. Cartan,  "Sur la structure des sous-ensembles analytiques réels"  ''C.R. Acad. Sci. Paris'' , '''244'''  (1957)  pp. 988–900</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Bruhat,  H. Cartan,  "Sur les composantes irréductibles d'un sous-ensemble"  ''C.R. Acad. Sci. Paris'' , '''244'''  (1957)  pp. 1123–1126</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F. Bruhat,  H. Whitney,  "Quelques propriétés fondamentales des ensembles analytiques-réels"  ''Comm. Math. Helv.'' , '''33'''  (1959)  pp. 132–160</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Narasimhan,  "Introduction to the theory of analytic spaces" , ''Lect. notes in math.'' , '''25''' , Springer  (1966)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Grauert,  R. Remmert,  "Theory of Stein spaces" , Springer  (1979)  (Translated from German)</TD></TR></table>

Revision as of 08:10, 6 June 2020


An analytic space over the field $ \mathbf R $ of real numbers. Unlike in the case of complex-analytic spaces, the structure sheaves of real-analytic spaces need not be coherent (cf. Coherent sheaf). Real-analytic spaces are said to be coherent if their structure sheaves are coherent. All real-analytic manifolds (i.e. smooth real-analytic spaces) are coherent real-analytic spaces.

Let $ V _ {a} $ be the germ at a point $ a $ of a real-analytic subset of $ \mathbf R ^ {n} $( cf. Analytic set). This defines the germ at $ a $ of a complex-analytic subset $ {\widetilde{V} } _ {a} $ of the space $ \mathbf C ^ {n} $ with the following equivalent properties: 1) $ \widetilde{V} _ {a} $ is the intersection of all germs of complex-analytic sets containing $ V _ {a} $; 2) if $ {\mathcal O} _ {V _ {a} } $ is the analytic algebra of the germ $ V _ {a} $, then $ {\mathcal O} _ {V _ {a} } \otimes \mathbf C $ is the analytic algebra of the germ $ \widetilde{V} _ {a} $. The germ $ \widetilde{V} _ {a} $ is said to be the complexification of the germ $ V _ {a} $, and $ V _ {a} $ is said to be the real part of the germ $ \widetilde{V} _ {a} $. Analogously, for any coherent real-analytic countably-infinite space $ X $ it is possible to construct the complexification $ \widetilde{X} $, which is a complex-analytic space. $ X $ will then have a fundamental system of neighbourhoods in $ \widetilde{X} $ which are Stein spaces (cf. Stein space).

The theory of coherent real-analytic spaces is similar to the theory of complex Stein spaces. Global sections of any coherent analytic sheaf of modules $ F $ on a coherent real-analytic countably-infinite space $ X $ generate modules of germs of its sections at any point of $ X $, and all groups $ H ^ {q} ( X, F ) $ vanish if $ q \geq 1 $.

For any finite-dimensional coherent real-analytic countably-infinite space $ ( X, {\mathcal O} _ {X} ) $ there exists a morphism

$$ f = ( f _ {0} , f _ {1} ): ( X, {\mathcal O} _ {X} ) \rightarrow \ ( \mathbf R ^ {n} , {\mathcal O} _ {\mathbf R ^ {n} } ) $$

such that $ f _ {0} $ is a proper one-to-one mapping of $ X $ into a coherent subspace in $ \mathbf R ^ {n} $, while $ f $ is an imbedding at the smooth points of $ X $. In particular, any (Hausdorff and countably-infinite) real-analytic manifold is isomorphic to a real-analytic submanifold in $ \mathbf R ^ {n} $. For a reduced coherent real-analytic space $ X $ the set of classes of isomorphic real-analytic principal fibre bundles with a real structure Lie group $ G $, admitting complexification, and base $ X $ is in one-to-one correspondence with the set of classes of isomorphic topological principal fibre bundles with the same structure group $ G $.

References

[1] A. Tognoli, "Some results in the theory of real analytic spaces" M. Jurchesan (ed.) , Espaces Analytiques (Bucharest 1969) , Acad. Roumanie (1971) pp. 149–157

Comments

References

[a1] H. Cartan, "Variétés analytiques réelles et variétés analytiques complexes" Bull. Soc. Math. France , 85 (1957) pp. 77–99
[a2] F. Bruhat, H. Cartan, "Sur la structure des sous-ensembles analytiques réels" C.R. Acad. Sci. Paris , 244 (1957) pp. 988–900
[a3] F. Bruhat, H. Cartan, "Sur les composantes irréductibles d'un sous-ensemble" C.R. Acad. Sci. Paris , 244 (1957) pp. 1123–1126
[a4] F. Bruhat, H. Whitney, "Quelques propriétés fondamentales des ensembles analytiques-réels" Comm. Math. Helv. , 33 (1959) pp. 132–160
[a5] R. Narasimhan, "Introduction to the theory of analytic spaces" , Lect. notes in math. , 25 , Springer (1966)
[a6] H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1979) (Translated from German)
How to Cite This Entry:
Real-analytic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Real-analytic_space&oldid=14527
This article was adapted from an original article by D.A. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article