View source for Nash theorems (in differential geometry)

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" />.

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" />.

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====
<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>



====Comments====
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" />.

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" />.

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" />.

====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>