RdL

Isometric Immersions

Keywords: Nash Theorem, Isometric Immersions, Underdetermined PDE systems, Free Maps, Implicit Function Theorem

In his book Partial Differential Relations, M. Gromov proves, among a wealth of many other results, the following infinite-dimensional version of the Implicit Function Theorem, inspired by the seminal works of John Nash on the existence of isometric immersions.

Definition. Let $M$ be a $n$-dimensional manifold, $\hbox{Imm}^1(M,\mathbb R^q)$ the set of $C^1$ immersions of $M$ into $\mathbb R^q$, ${\cal G}^0(M)$ the bundle of $C^0$ Riemannian metrics over $M$ and $e_q$ the Euclidean metric on $\mathbb R^q$. Let ${\cal D}_{M,q}:\hbox{Imm}^1(M,\mathbb R^q)\to{\cal G}^0(M)$ be the metric-inducing operator defined by $$ {\cal D}_{M,q}(f) = f^*e_q. $$ Let $\Gamma^0\left(S^0_2(M)\right)$ be the set of $C^0$ sections of the tensor bundle of symmetric $(0,2)$ tensors over $M$. The Partial Differential Operator $$ T_{f_0}{\cal D}_{M,q}:T_{f_0}C^1(M,\mathbb R^q)\simeq C^1(M,\mathbb R^q)\to T_{g_0}{\cal G}^0(M)\simeq\Gamma^0\left(S^0_2(M)\right) $$ defined by $$ T_{f_0}{\cal D}_{M,q}(\delta f)=2\delta_{ij}\;\partial_\alpha f_0^i\;\partial_\beta \delta f^j $$ is the tangent map (linearization) of ${\cal D}_{M,q}$ at $f_0$. We say that ${\cal D}_{M,q}$ is infinitesimally invertible over ${\cal A}\subset C^1(M,\mathbb R^q)$ if there exists a family ${\cal E}$ of linear PDOs ${\cal E}_f:\Gamma^s(S^0_2M)\to T_fC^0(M,\mathbb R^q)$, $f\in{\cal A}$, of some order $s$ such that:
  1. there is an integer $d\geq1$, called {\em defect} of ${\cal E}$, such that ${\cal A}={\cal A}^d\subset C^d(M,\mathbb R^q)$ and ${\cal A}$ is defined by some open differential relation;
  2. the map ${\cal E}:{\cal A}^d\times\Gamma^s(S^0_2M)\to TC^0(M,\mathbb R^q)$ defined by ${\cal E}(f,\eta)\to{\cal E}_f(\eta)$ is a PDO of order $d$ in the first variable and order $s$ in the second;
  3. $T_{f}{\cal D}_{M,q}({\cal E}_{f}(\delta g))=\delta g$ for all $f\in{\cal A}^d\cap C^{d+1}(M,\mathbb R^q)$ and $\delta g\in\Gamma^{s+1}(S^0_2M)$.

Theorem [Nash-Gromov IFT, 1986]. Let $F\to E$ be a fiber bundle, $G\to E$ a vector bundle and $\Gamma^rF$ and $\Gamma^sG$, respectively, the sets of their $C^r$ and $C^s$ sections. If a PDO of order $r$, ${\cal D}_r:\Gamma^rF\to\Gamma^0G$, admits an infinitesimal inversion of order $s$ and defect $d$ over ${\cal A}^d\subset\Gamma^dF$, then for every $f_0\in{\cal A}^d\cap\Gamma^\infty F$ there exists a neighborhood of zero ${\cal U}\subset \Gamma^{\bar s+s+1}G$, $\bar s=\max\{d,2r+s\}$, such that, for every $g\in{\cal U}\cap\Gamma^{\sigma+s}G$, $\sigma\geq \bar s+1$, the PDE $${\cal D}(f)={\cal D}(f_0)+g$$ has a $C^\sigma$ solution.

Corollary [Gromov, 1986]. Under the hypotheses of the theorem above, the restriction of ${\cal D}_r$ to ${\cal A}^\infty={\cal A}^d\cap\Gamma^\infty E$ is an open map.

As a corollary of this general theorem, taking $\cal A^\infty$ as the set of Free Maps $f:M\mathbb\to\mathbb R^q$, namely those maps whose first and second partial derivatives are linearly independent at every point, one gets immediately the following celebrated result of Nash:

Theorem [Nash, 1956]. The restriction of ${\cal D}_{M,q}$ to $\hbox{Free}^\infty(M,\mathbb R^q)$ is an open map.

Jointly with G. D'Ambra and A. Loi, we used the Nash-Gromov's IFT above to extend the result of Nash to "partial isometries" as illustrated below.

Definition. Given a vector subbundle ${\cal H}\subset TM$, we say that $f\in C^\infty(M,\mathbb R^q)$ is a $\cal H$-immersion if $Tf|_{\cal H}$ is injective and that is $\cal H$-free if, given a local trivialization $\{\xi_1,\dots,\xi_k\}$ of $\cal H$, so that $\cal H=\hbox{span}\{\xi_1,\dots,\xi_k\}$ over some $U\subset M$, the matrix of the first and symmetrized second Lie derivatives of $f$ with respect to the $\xi_i$ is non-degenerate at every point of $U$. We denote by $\cal D_{\cal H,q}$ the operator defined by $\cal D_{\cal H,q}(f)=(f^*e_q)|_{\cal H}$.
Theorem [D'Ambra, De Leo, Loi, 2010.] The restriction of ${\cal D}_{\cal H,q}$ to ${\cal H}$-$\hbox{Free}^\infty(M,\mathbb R^q)$ is an open map.

There are three important cases that I devised where this theorem applies:

1. One-dimensional distributions of finite type on the plane.

2. Lagrangian distributions.

3. One-dimensional Riemann-Poisson distributions. More more detsails, see

I also investigated a second type of problem. Free maps from $M^n$ to $\mathbb R^q$ can arise only when $q\geq n+n(n+1)/2$, so when $q$ is lower than that value Nash theorem is empty. Do isometric immersions arise also for lower values of $q$? And how low can $q$ be? In Sec. 2.3.8 (E) of his monograph, Gromov formulated the following:

Conjecture [Gromov, 1986]. If $q>n(n+1)/2$, the operator ${\cal D}_{M,q}$ is open over an open dense subset of smooth maps.

Using a technique suggested by Gromov in his book, I was able to prove the following:

Theorem [De Leo, 2017]. If $q\geq n+n(n+1)/2-\sqrt{n/2}+1/2$, the operator ${\cal D}_{M,q}$ is open over a non-empty open subset of smooth maps.

For more results and examples, see

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