# Nash theorems (in differential geometry)

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 , and in the form stated above in . 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 ; 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 ). 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 ).