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In every analytic [[Riemannian manifold|Riemannian manifold]] of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054190/j0541901.png" /> there exists a neighbourhood of an arbitrarily chosen point having an isometric imbedding into the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054190/j0541902.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054190/j0541903.png" />. Janet's theorem remains true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054190/j0541904.png" /> is replaced by any analytic Riemannian manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054190/j0541905.png" /> with a prescribed point (to which the point chosen in the original manifold must be mapped). Janet's theorem is valid in the case of pseudo-Riemannian manifolds provided that
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<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/j/j054/j054190/j0541906.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054190/j0541907.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054190/j0541908.png" /> are the dimensions of the positive and negative parts of the metric tensor on the original manifold, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054190/j0541909.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054190/j05419010.png" /> are the corresponding dimensions of the target manifold (see [[#References|[3]]]). Janet's theorem is the first general imbedding theorem in Riemannian geometry (see [[Isometric immersion|Isometric immersion]]).
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In every analytic [[Riemannian manifold|Riemannian manifold]] of dimension  $  n $
 +
there exists a neighbourhood of an arbitrarily chosen point having an isometric imbedding into the Euclidean space  $  \mathbf R ^ {s _ {n} } $
 +
of dimension  $  s _ {n} = n ( n + 1 ) / 2 $.  
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Janet's theorem remains true if  $  \mathbf R ^ {s _ {n} } $
 +
is replaced by any analytic Riemannian manifold of dimension  $  s _ {n} $
 +
with a prescribed point (to which the point chosen in the original manifold must be mapped). Janet's theorem is valid in the case of pseudo-Riemannian manifolds provided that
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$$
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q  \geq  s _ {n} ,\  q _ {+}  \geq  n _ {+} ,\ \
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q _ {-}  \geq  n _ {-} ,
 +
$$
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 +
where  $  n _ {+} $
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and $  n _ {-} = n - n _ {+} $
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are the dimensions of the positive and negative parts of the metric tensor on the original manifold, and $  q _ {+} $
 +
and $  q _ {-} = q - q _ {+} $
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are the corresponding dimensions of the target manifold (see [[#References|[3]]]). Janet's theorem is the first general imbedding theorem in Riemannian geometry (see [[Isometric immersion|Isometric immersion]]).
  
 
Janet's theorem first appeared as a conjecture of L. Schläfli [[#References|[1]]], and was proved by M. Janet [[#References|[2]]].
 
Janet's theorem first appeared as a conjecture of L. Schläfli [[#References|[1]]], and was proved by M. Janet [[#References|[2]]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Schläfli,  "Nota alla Memoria del signor Beltrami  "Sugli spazi di curvatura costante" "  ''Ann. Mat. Pura. Appl. Ser. 2'' , '''5'''  (1873)  pp. 178–193</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Janet,  "Sur la possibilité de plonger un espace Riemannien donné dans un espace euclidien"  ''Ann. Soc. Polon. Math.'' , '''5'''  (1926)  pp. 38–43</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Friedman,  "Isometric imbedding of Riemannian manifolds into Euclidean spaces"  ''Rev. Modern Physics'' , '''77'''  (1965)  pp. 201–203</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Schläfli,  "Nota alla Memoria del signor Beltrami  "Sugli spazi di curvatura costante" "  ''Ann. Mat. Pura. Appl. Ser. 2'' , '''5'''  (1873)  pp. 178–193</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Janet,  "Sur la possibilité de plonger un espace Riemannien donné dans un espace euclidien"  ''Ann. Soc. Polon. Math.'' , '''5'''  (1926)  pp. 38–43</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Friedman,  "Isometric imbedding of Riemannian manifolds into Euclidean spaces"  ''Rev. Modern Physics'' , '''77'''  (1965)  pp. 201–203</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 22:14, 5 June 2020


In every analytic Riemannian manifold of dimension $ n $ there exists a neighbourhood of an arbitrarily chosen point having an isometric imbedding into the Euclidean space $ \mathbf R ^ {s _ {n} } $ of dimension $ s _ {n} = n ( n + 1 ) / 2 $. Janet's theorem remains true if $ \mathbf R ^ {s _ {n} } $ is replaced by any analytic Riemannian manifold of dimension $ s _ {n} $ with a prescribed point (to which the point chosen in the original manifold must be mapped). Janet's theorem is valid in the case of pseudo-Riemannian manifolds provided that

$$ q \geq s _ {n} ,\ q _ {+} \geq n _ {+} ,\ \ q _ {-} \geq n _ {-} , $$

where $ n _ {+} $ and $ n _ {-} = n - n _ {+} $ are the dimensions of the positive and negative parts of the metric tensor on the original manifold, and $ q _ {+} $ and $ q _ {-} = q - q _ {+} $ are the corresponding dimensions of the target manifold (see [3]). Janet's theorem is the first general imbedding theorem in Riemannian geometry (see Isometric immersion).

Janet's theorem first appeared as a conjecture of L. Schläfli [1], and was proved by M. Janet [2].

References

[1] L. Schläfli, "Nota alla Memoria del signor Beltrami "Sugli spazi di curvatura costante" " Ann. Mat. Pura. Appl. Ser. 2 , 5 (1873) pp. 178–193
[2] M. Janet, "Sur la possibilité de plonger un espace Riemannien donné dans un espace euclidien" Ann. Soc. Polon. Math. , 5 (1926) pp. 38–43
[3] A. Friedman, "Isometric imbedding of Riemannian manifolds into Euclidean spaces" Rev. Modern Physics , 77 (1965) pp. 201–203

Comments

The theorem was also proved by E. Cartan [a1]. A rigorous proof along the lines suggested by Janet was given by C. Burstin [a2]. See also [a3].

References

[a1] E. Cartan, "Sur la possibilité de plonger un espace riemannien donné dans un espace euclidéen" Ann. Soc. Polon. Math. , 6 (1927) pp. 1–7
[a2] C. Burstin, Mat. Sb. , 38 (1931) pp. 74–93
[a3] M. Spivak, "A comprehensive introduction to differential geometry" , 5 , Publish or Perish (1975) pp. 1–5
How to Cite This Entry:
Janet theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Janet_theorem&oldid=47462
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article