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; Isometric immersion). The original versions are due to J. Nash [1].
1) Nash' theorem on -imbeddings and
-immersions. An immersion (imbedding)
of class
of an
-dimensional Riemannian space
with metric
of class
into an
-dimensional Euclidean space
is called short if the metric
induced by it on
is such that the quadratic form
is positive definite. If
has a short immersion (imbedding) in
,
, then
also has an isometric immersion (imbedding) of class
in
. Under the restriction
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
has a
-imbedding (immersion) in
,
, then
also has an isometric
-imbedding (immersion) in
. Another consequence of Nash' theorem is that every point of
has a sufficiently small neighbourhood that admits an isometric imbedding of class
in
.
2) Nash' theorem on regular imbeddings. Every compact Riemannian manifold of class
,
, has an isometric
-imbedding in
, where
. If
is not compact, then it has an isometric
-imbedding in
, where
.
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 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,
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
with respect to the intrinsic coordinates of
must be linearly independent. Such imbeddings were first considered in [4]; they are called free. Nash' implicit-function theorem implies that a compact Riemannian manifold
sufficiently close to another one
having a free imbedding in
also has a free imbedding in
. This fact and the original method of extension with respect to a parameter lead to Nash' theorem on regular imbeddings (see [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
has an isometric differentiable (analytic) imbedding in
, where
(see [5]–[7]).
References
[1] | J. Nash, "![]() |
[2] | N. Kuiper, "On ![]() |
[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 -manifold without boundary is diffeomorphic to a component of a real algebraic variety.
Let be a smooth (i.e.
-) fibration. Denote by
the space of
-jets (of germs) of smooth sections
(cf. Germ; Jet). The
-th order jet of a section
is denoted by
. A section
is called holonomic if there is a
-section
such that
;
determines
uniquely (if it exists). The fine topology on the space
of
-sections
is obtained by taking as a basis the subsets
where
runs over the open subsets of
. The fine
-topology on
is induced by the imbedding
,
, from the fine
-topology to
.
The Nash approximation theorem says that an arbitrary Riemannian -metric on
has a fine
-approximation by some
-metric
on
that admits
-immersions
for some
, where
.
The Nash–Kuiper theorem [1], [2] says that an arbitrary differentiable immersion for
admits a
-continuous homotopy of immersions
,
, to an isometric immersion
.
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 |
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