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Two groups of theorems on isometrically imbedded and immersed Riemannian manifolds in a Euclidean space (see also [[Immersion of a manifold|Immersion of a manifold]]; [[Isometric immersion|Isometric immersion]]). The original versions are due to J. Nash [[#References|[1]]].
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Two groups of theorems on isometrically imbedded and immersed Riemannian manifolds in a Euclidean space (see also [[Immersion of a manifold|Immersion of a manifold]]; [[Isometric immersion|Isometric immersion]]). The original versions are due to J. Nash ([[#References|[1]]]).
  
1) Nash' theorem on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n0660503.png" />-imbeddings and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n0660504.png" />-immersions. An immersion (imbedding) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n0660505.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n0660506.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n0660507.png" />-dimensional [[Riemannian space|Riemannian space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n0660508.png" /> with metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n0660509.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605010.png" /> into an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605011.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605012.png" /> is called short if the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605013.png" /> induced by it on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605014.png" /> is such that the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605015.png" /> is positive definite. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605016.png" /> has a short immersion (imbedding) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605018.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605019.png" /> also has an isometric immersion (imbedding) of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605021.png" />. Under the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605022.png" /> this theorem was proved in [[#References|[1]]], and in the form stated above in [[#References|[2]]]. This theorem implies, in particular, that if a compact Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605023.png" /> has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605024.png" />-imbedding (immersion) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605027.png" /> also has an isometric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605028.png" />-imbedding (immersion) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605029.png" />. Another consequence of Nash' theorem is that every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605030.png" /> has a sufficiently small neighbourhood that admits an isometric imbedding of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605032.png" />.
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# '''Nash’s theorem on $ C^{1} $-imbeddings and $ C^{1} $-immersions.''' A $ C^{1} $-immersion (-imbedding) $ f: V^{n} \to E^{m} $ of an $ n $-dimensional [[Riemannian space|Riemannian space]] $ V^{n} $ with a $ C^{0} $-metric $ g $ into an $ m $-dimensional Euclidean space $ E^{m} $ is called '''short''' if and only if the metric $ g_{f} $ induced by it on $ V^{n} $ is such that the quadratic form $ g - g_{f} $ is positive definite. If $ V^{n} $ has a short immersion (imbedding) into $ E^{m} $, where $ m \geq n + 1 $, then $ V^{n} $ also has an isometric $ C^{1} $-immersion (-imbedding) into $ E^{m} $. Under the restriction $ m \geq n + 2 $, this theorem was proved in [[#References|[1]]], and in the form stated above in [[#References|[2]]]. This theorem implies, in particular, that if a compact Riemannian manifold $ V^{n} $ has a $ C^{1} $-imbedding (-immersion) into $ E^{m} $, where $ m \geq n + 1 $, then $ V^{n} $ also has an isometric $ C^{1} $-imbedding (-immersion) into $ E^{m} $. Another consequence of Nash’s theorem is that every point of $ V^{n} $ has a sufficiently small neighborhood that admits an isometric $ C^{1} $-imbedding into $ E^{n + 1} $.
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# '''Nash’s theorem on regular imbeddings.''' Every compact Riemannian manifold $ V^{n} $ of class $ C^{r} $, where $ 3 \leq r \leq \infty $, has an isometric $ C^{r} $-imbedding into $ E^{m} $, where $ m = \dfrac{3 n^{2} + 11 n}{2} $. If $ V^{n} $ is not compact, then it has an isometric $ C^{r} $-imbedding into $ E^{m_{1}} $, where $ m_{1} = \dfrac{(3 n^{2} + 11) (n + 1)}{2} $.
  
2) Nash' theorem on regular imbeddings. Every compact Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605033.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605035.png" />, has an isometric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605036.png" />-imbedding in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605038.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605039.png" /> is not compact, then it has an isometric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605040.png" />-imbedding in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605041.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605042.png" />.
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Nash’s theorem on regular imbeddings results from an application of Nash’s implicit-function theorem on the inversion of a broad class of differential operators. The meaning of this theorem is that when a certain linear algebraic system of equations connected naturally with a differential operator $ L $ is solvable, and when a reasonable topology is introduced in the image and inverse image, then the operator in question is an open mapping, i.e., $ L $ is locally invertible near any point of its range. For the equations of an imbedding of a Riemannian manifold into a Euclidean space, this reduces to the fact that the first and second derivatives of the mapping $ f: V^{n} \to E^{m} $ with respect to the intrinsic coordinates of $ V^{n} $ must be linearly independent. Such imbeddings were first considered in [[#References|[4]]]; they are called '''free'''. Nash’s implicit-function theorem implies that a compact Riemannian manifold $ V^{n} $ that is sufficiently close to another one $ W^{n} $ having a free imbedding into $ E^{m} $ also has a free imbedding into $ E^{m} $. This fact, and the original method of extension with respect to a parameter, lead to Nash’s theorem on regular imbeddings (see [[#References|[3]]]). By extending Nash’s method to non-compact manifolds and analytic imbeddings, and also by a principal refinement of the process of extension with respect to a parameter, it has been proved that every infinitely differentiable (analytic) Riemannian manifold $ V^{n} $ has an isometric differentiable (analytic) imbedding into $ E^{m} $, where $ m = \dfrac{n (n + 1)}{2} + 3 n + 5 $ (see [[#References|[5]]]–[[#References|[7]]]).
 
 
Nash' theorem on regular imbeddings results from an application of Nash' implicit-function theorem on the inversion of a broad class of differential operators. The meaning of this theorem is that when a certain linear algebraic system of equations connected naturally with a differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605043.png" /> is solvable and when a reasonable topology is introduced in the image and inverse image, then the operator in question is an open mapping, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605044.png" /> is locally invertible near any point of its range. For the equations of an imbedding of a Riemannian manifold in a Euclidean space this reduces to the fact that the first and second derivatives of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605045.png" /> with respect to the intrinsic coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605046.png" /> must be linearly independent. Such imbeddings were first considered in [[#References|[4]]]; they are called free. Nash' implicit-function theorem implies that a compact Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605047.png" /> sufficiently close to another one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605048.png" /> having a free imbedding in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605049.png" /> also has a free imbedding in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605050.png" />. This fact and the original method of extension with respect to a parameter lead to Nash' theorem on regular imbeddings (see [[#References|[3]]]). By extending Nash' method to non-compact manifolds and analytic imbeddings, and also by a principal refinement of the process of extension with respect to a parameter, it has been proved that every infinitely-differentiable (analytic) Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605051.png" /> has an isometric differentiable (analytic) imbedding in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605053.png" /> (see [[#References|[5]]]–[[#References|[7]]]).
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Nash,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605054.png" />-isometric imbeddings"  ''Ann. of Math.'' , '''60'''  (1954)  pp. 383–396  {{MR|0065993}} {{ZBL|0058.37703}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Kuiper,  "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605055.png" />-isometric imbeddings"  ''Proc. K. Ned. Akad. Wetensch.'' , '''A58''' :  4  (1955)  pp. 545–556  {{MR|75640}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Nash,  "The imbedding problem for Riemannian manifolds"  ''Ann. of Math.'' , '''63'''  (1956)  pp. 20–63  {{MR|0075639}} {{ZBL|0070.38603}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  C. Burstin,  "Ein Beitrag zum Problem der Einbettung der Riemannschen Räume in euklidischen Räumen"  ''Mat. Sb.'' , '''38''' :  3–4  (1931)  pp. 74–85  {{MR|}} {{ZBL|0006.08004}}  {{ZBL|57.0549.01}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Nash,  "Analyticity of the solutions of implicit function problems with analytic data"  ''Ann. of Math.'' , '''84'''  (1966)  pp. 345–355  {{MR|0205266}} {{ZBL|0173.09202}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M.L. Gromov,  V.A. Rokhlin,  "Embeddings and immersions in Riemannian geometry"  ''Russian Math. Surveys'' , '''25''' :  5  (1970)  pp. 1–57  ''Uspekhi Mat. Nauk'' , '''25'''  (1970)  pp. 53–62  {{MR|}} {{ZBL|0222.53053}} {{ZBL|0202.21004}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  M.L. Gromov,  "Isometric imbeddings and immersions"  ''Soviet Math. Dokl.'' , '''11''' :  3  (1970)  pp. 1206–1209  ''Dokl. Akad. Nauk SSSR'' , '''192'''  (1970)  pp. 794–797  {{MR|0275456}} {{ZBL|0214.50404}} </TD></TR></table>
 
  
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<table>
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<TR><TD valign="top">[1]</TD><TD valign="top">
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J. Nash, “$ C^{1} $-isometric imbeddings”, ''Ann. of Math.'', '''60''' (1954), pp. 383–396. {{MR|0065993}} {{ZBL|0058.37703}}</TD></TR>
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<TR><TD valign="top">[2]</TD><TD valign="top">
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N. Kuiper, “On $ C^{1} $-isometric imbeddings”, ''Proc. K. Ned. Akad. Wetensch.'', '''A58''': 4 (1955), pp. 545–556. {{MR|75640}}</TD></TR>
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<TR><TD valign="top">[3]</TD><TD valign="top">
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J. Nash, “The imbedding problem for Riemannian manifolds”, ''Ann. of Math.'', '''63''' (1956), pp. 20–63. {{MR|0075639}} {{ZBL|0070.38603}}</TD></TR>
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<TR><TD valign="top">[4]</TD><TD valign="top">
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C. Burstin, “Ein Beitrag zum Problem der Einbettung der Riemannschen Räume in euklidischen Räumen”, ''Mat. Sb.'', '''38''': 3–4 (1931), pp. 74–85. {{ZBL|0006.08004}} {{ZBL|57.0549.01}}</TD></TR>
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<TR><TD valign="top">[5]</TD><TD valign="top">
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J. Nash, “Analyticity of the solutions of implicit function problems with analytic data”, ''Ann. of Math.'', '''84''' (1966), pp. 345–355. {{MR|0205266}} {{ZBL|0173.09202}}</TD></TR>
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<TR><TD valign="top">[6]</TD><TD valign="top">
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M.L. Gromov, V.A. Rokhlin, “Embeddings and immersions in Riemannian geometry”, ''Russian Math. Surveys'', '''25''': 5 (1970), pp. 1–57; ''Uspekhi Mat. Nauk'', '''25''' (1970), pp. 53–62. {{ZBL|0222.53053}} {{ZBL|0202.21004}}</TD></TR>
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<TR><TD valign="top">[7]</TD><TD valign="top">
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M.L. Gromov, “Isometric imbeddings and immersions”, ''Soviet Math. Dokl.'', '''11''': 3 (1970), pp. 1206–1209; ''Dokl. Akad. Nauk SSSR'', '''192''' (1970), pp. 794–797. {{MR|0275456}} {{ZBL|0214.50404}}</TD></TR>
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</table>
  
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====Comments====
  
====Comments====
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The Nash theorem in differential topology says that a compact connected $ C^{\infty} $-manifold without boundary is diffeomorphic to a component of a real algebraic variety.
The Nash theorem in differential topology says that a compact connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605056.png" />-manifold without boundary is diffeomorphic to a component of a real algebraic variety.
 
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605057.png" /> be a smooth (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605058.png" />-) fibration. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605059.png" /> the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605060.png" />-jets (of germs) of smooth sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605061.png" /> (cf. [[Germ|Germ]]; [[Jet|Jet]]). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605062.png" />-th order jet of a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605063.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605064.png" />. A section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605065.png" /> is called holonomic if there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605066.png" />-section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605067.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605068.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605069.png" /> determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605070.png" /> uniquely (if it exists). The fine topology on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605071.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605072.png" />-sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605073.png" /> is obtained by taking as a basis the subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605074.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605075.png" /> runs over the open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605076.png" />. The fine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605077.png" />-topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605078.png" /> is induced by the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605080.png" />, from the fine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605081.png" />-topology to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605082.png" />.
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Let $ \pi: X \to V $ be a smooth (i.e., $ C^{\infty} $-) fibration. Denote by $ {J^{r}}(V,X) $ the space of $ r $-[[Jet|jets]] (of [[Germ|germs]]) of smooth sections $ f: V \to X $. The $ r $-th order jet of a section $ f: V \to X $ is denoted by $ J^{r} f: V \to {J^{r}}(V,X) $. A section $ \phi: V \to {J^{r}}(V,X) $ is called '''holonomic''' if and only if there is a $ C^{r} $-section $ f: V \to X $ such that $ \phi = J^{r} f $; note that $ \phi $ determines $ f $ uniquely (if it exists). The fine topology on the space $ {C^{0}}(V,X) $ of $ C^{0} $-sections $ f: V \to X $ is obtained by taking as a basis the subsets $ {C^{0}}(V,U) $, where $ U $ runs over the open subsets of $ X $. The fine $ C^{r} $-topology on $ {C^{r}}(V,X) $ is induced by the imbedding $ \left\{ \begin{matrix} {C^{r}}(V,X) & \to & {C^{0}}(V,{J^{r}}(V,X)) \\ f & \mapsto & J^{r} f \end{matrix} \right\} $ from the fine $ C^{0} $-topology to $ {C^{0}}(V,{J^{r}}(V,X)) $.
  
The Nash approximation theorem says that an arbitrary Riemannian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605083.png" />-metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605084.png" /> has a fine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605085.png" />-approximation by some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605086.png" />-metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605087.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605088.png" /> that admits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605089.png" />-immersions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605090.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605091.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605092.png" />.
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The Nash approximation theorem says that an arbitrary Riemannian $ C^{r} $-metric $ g $ on $ V $ has a fine $ C^{r} $-approximation by some $ C^{r} $-metric $ g' $ on $ V $ that admits $ C^{r} $-immersions $ f': (V,g') \to \mathbf{R}^{2 l} $ for some $ l = l(n) < \infty $, where $ n = \dim(V) $.
  
The Nash–Kuiper theorem [[#References|[1]]], [[#References|[2]]] says that an arbitrary differentiable immersion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605093.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605094.png" /> admits a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605095.png" />-continuous homotopy of immersions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605097.png" />, to an isometric immersion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066050/n06605098.png" />.
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The Nash–Kuiper theorem ([[#References|[1]]], [[#References|[2]]]) says that an arbitrary differentiable immersion $ f_{0}: V \to \mathbf{R}^{q} $ for $ q > \dim(V) $ admits a $ C^{1} $-continuous homotopy $ (f_{t}: V \to \mathbf{R}^{q})_{t \in [0,1]} $ of immersions to an isometric immersion $ f_{1}: V \to \mathbf{R}^{q} $.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M.W. Hirsch,   "Differential topology" , Springer (1976) {{MR|0448362}} {{ZBL|0356.57001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Gromov,   "Partial differential relations" , Springer (1986) (Translated from Russian) {{MR|0864505}} {{ZBL|0651.53001}} </TD></TR></table>
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 +
<table>
 +
<TR><TD valign="top">[a1]</TD><TD valign="top">
 +
M.W. Hirsch, “Differential topology”, Springer (1976). {{MR|0448362}} {{ZBL|0356.57001}}</TD></TR>
 +
<TR><TD valign="top">[a2]</TD><TD valign="top">
 +
M. Gromov, “Partial differential relations”, Springer (1986). (Translated from Russian) {{MR|0864505}} {{ZBL|0651.53001}}</TD></TR>
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</table>

Latest revision as of 16:28, 16 January 2017

Two groups of theorems on isometrically imbedded and immersed Riemannian manifolds in a Euclidean space (see also Immersion of a manifold; Isometric immersion). The original versions are due to J. Nash ([1]).

  1. Nash’s theorem on $ C^{1} $-imbeddings and $ C^{1} $-immersions. A $ C^{1} $-immersion (-imbedding) $ f: V^{n} \to E^{m} $ of an $ n $-dimensional Riemannian space $ V^{n} $ with a $ C^{0} $-metric $ g $ into an $ m $-dimensional Euclidean space $ E^{m} $ is called short if and only if the metric $ g_{f} $ induced by it on $ V^{n} $ is such that the quadratic form $ g - g_{f} $ is positive definite. If $ V^{n} $ has a short immersion (imbedding) into $ E^{m} $, where $ m \geq n + 1 $, then $ V^{n} $ also has an isometric $ C^{1} $-immersion (-imbedding) into $ E^{m} $. Under the restriction $ m \geq n + 2 $, this theorem was proved in [1], and in the form stated above in [2]. This theorem implies, in particular, that if a compact Riemannian manifold $ V^{n} $ has a $ C^{1} $-imbedding (-immersion) into $ E^{m} $, where $ m \geq n + 1 $, then $ V^{n} $ also has an isometric $ C^{1} $-imbedding (-immersion) into $ E^{m} $. Another consequence of Nash’s theorem is that every point of $ V^{n} $ has a sufficiently small neighborhood that admits an isometric $ C^{1} $-imbedding into $ E^{n + 1} $.
  2. Nash’s theorem on regular imbeddings. Every compact Riemannian manifold $ V^{n} $ of class $ C^{r} $, where $ 3 \leq r \leq \infty $, has an isometric $ C^{r} $-imbedding into $ E^{m} $, where $ m = \dfrac{3 n^{2} + 11 n}{2} $. If $ V^{n} $ is not compact, then it has an isometric $ C^{r} $-imbedding into $ E^{m_{1}} $, where $ m_{1} = \dfrac{(3 n^{2} + 11) (n + 1)}{2} $.

Nash’s theorem on regular imbeddings results from an application of Nash’s implicit-function theorem on the inversion of a broad class of differential operators. The meaning of this theorem is that when a certain linear algebraic system of equations connected naturally with a differential operator $ L $ is solvable, and when a reasonable topology is introduced in the image and inverse image, then the operator in question is an open mapping, i.e., $ L $ is locally invertible near any point of its range. For the equations of an imbedding of a Riemannian manifold into a Euclidean space, this reduces to the fact that the first and second derivatives of the mapping $ f: V^{n} \to E^{m} $ with respect to the intrinsic coordinates of $ V^{n} $ must be linearly independent. Such imbeddings were first considered in [4]; they are called free. Nash’s implicit-function theorem implies that a compact Riemannian manifold $ V^{n} $ that is sufficiently close to another one $ W^{n} $ having a free imbedding into $ E^{m} $ also has a free imbedding into $ E^{m} $. This fact, and the original method of extension with respect to a parameter, lead to Nash’s theorem on regular imbeddings (see [3]). By extending Nash’s method to non-compact manifolds and analytic imbeddings, and also by a principal refinement of the process of extension with respect to a parameter, it has been proved that every infinitely differentiable (analytic) Riemannian manifold $ V^{n} $ has an isometric differentiable (analytic) imbedding into $ E^{m} $, where $ m = \dfrac{n (n + 1)}{2} + 3 n + 5 $ (see [5][7]).

References

[1] J. Nash, “$ C^{1} $-isometric imbeddings”, Ann. of Math., 60 (1954), pp. 383–396. MR0065993 Zbl 0058.37703
[2] N. Kuiper, “On $ C^{1} $-isometric imbeddings”, Proc. K. Ned. Akad. Wetensch., A58: 4 (1955), pp. 545–556. MR75640
[3] J. Nash, “The imbedding problem for Riemannian manifolds”, Ann. of Math., 63 (1956), pp. 20–63. MR0075639 Zbl 0070.38603
[4] C. Burstin, “Ein Beitrag zum Problem der Einbettung der Riemannschen Räume in euklidischen Räumen”, Mat. Sb., 38: 3–4 (1931), pp. 74–85. Zbl 0006.08004 Zbl 57.0549.01
[5] J. Nash, “Analyticity of the solutions of implicit function problems with analytic data”, Ann. of Math., 84 (1966), pp. 345–355. MR0205266 Zbl 0173.09202
[6] M.L. Gromov, V.A. Rokhlin, “Embeddings and immersions in Riemannian geometry”, Russian Math. Surveys, 25: 5 (1970), pp. 1–57; Uspekhi Mat. Nauk, 25 (1970), pp. 53–62. Zbl 0222.53053 Zbl 0202.21004
[7] M.L. Gromov, “Isometric imbeddings and immersions”, Soviet Math. Dokl., 11: 3 (1970), pp. 1206–1209; Dokl. Akad. Nauk SSSR, 192 (1970), pp. 794–797. MR0275456 Zbl 0214.50404

Comments

The Nash theorem in differential topology says that a compact connected $ C^{\infty} $-manifold without boundary is diffeomorphic to a component of a real algebraic variety.

Let $ \pi: X \to V $ be a smooth (i.e., $ C^{\infty} $-) fibration. Denote by $ {J^{r}}(V,X) $ the space of $ r $-jets (of germs) of smooth sections $ f: V \to X $. The $ r $-th order jet of a section $ f: V \to X $ is denoted by $ J^{r} f: V \to {J^{r}}(V,X) $. A section $ \phi: V \to {J^{r}}(V,X) $ is called holonomic if and only if there is a $ C^{r} $-section $ f: V \to X $ such that $ \phi = J^{r} f $; note that $ \phi $ determines $ f $ uniquely (if it exists). The fine topology on the space $ {C^{0}}(V,X) $ of $ C^{0} $-sections $ f: V \to X $ is obtained by taking as a basis the subsets $ {C^{0}}(V,U) $, where $ U $ runs over the open subsets of $ X $. The fine $ C^{r} $-topology on $ {C^{r}}(V,X) $ is induced by the imbedding $ \left\{ \begin{matrix} {C^{r}}(V,X) & \to & {C^{0}}(V,{J^{r}}(V,X)) \\ f & \mapsto & J^{r} f \end{matrix} \right\} $ from the fine $ C^{0} $-topology to $ {C^{0}}(V,{J^{r}}(V,X)) $.

The Nash approximation theorem says that an arbitrary Riemannian $ C^{r} $-metric $ g $ on $ V $ has a fine $ C^{r} $-approximation by some $ C^{r} $-metric $ g' $ on $ V $ that admits $ C^{r} $-immersions $ f': (V,g') \to \mathbf{R}^{2 l} $ for some $ l = l(n) < \infty $, where $ n = \dim(V) $.

The Nash–Kuiper theorem ([1], [2]) says that an arbitrary differentiable immersion $ f_{0}: V \to \mathbf{R}^{q} $ for $ q > \dim(V) $ admits a $ C^{1} $-continuous homotopy $ (f_{t}: V \to \mathbf{R}^{q})_{t \in [0,1]} $ of immersions to an isometric immersion $ f_{1}: V \to \mathbf{R}^{q} $.

References

[a1] M.W. Hirsch, “Differential topology”, Springer (1976). MR0448362 Zbl 0356.57001
[a2] M. Gromov, “Partial differential relations”, Springer (1986). (Translated from Russian) MR0864505 Zbl 0651.53001
How to Cite This Entry:
Nash theorems (in differential geometry). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nash_theorems_(in_differential_geometry)&oldid=40187
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article