## Characteristics and existence of isometric embeddings.(English)Zbl 0536.53022

The paper considers an n-dimensional manifold $$(M^ n$$, $$ds^ 2)$$, as well as an exterior system $$(I,\omega)$$, whose integrals give local embeddings $$(M^ n,ds^ 2)$$ into $$E^{n(n-1)/2}$$. The basic invariant of this system is the characteristic sheaf $${\mathcal M}$$, which can be considered as a family of vector spaces $$M_{(x,\xi)}$$ of varying dimensions having $$\sup p {\mathcal M}=\{(x,\xi): \dim {\mathcal M}_{(x,\xi)}>0\}$$ and which is the characteristic variety $$\Xi$$. The authors prove that there exists a rational involution J, canonically defined on $$\Xi$$, and that $${\mathcal M}$$ is uniquely determined by ($$\Xi$$,J). The characteristic $$\Xi$$ is the union $$\Xi =\cup_{x\in M}\Xi_ x$$ of projective algebraic varieties $$\Xi_ x\in P_{n-1}$$ described in detail in the paper. For the isometric embedding system, the symbol map of the linear variational equations of the integral manifold N induced by $$(I,\omega)$$ is given by $$\gamma_{ij}:\quad W\otimes S^ 2v^ x\to K,$$ where $$W\cong R^{n(n-1)/2}$$ is the normal space, $$V\cong R^ n$$ is the cotangent space and $$K\cong R^{n^ 2(n^ 2-1)/12}$$ is the curvature-like tensor space. If one represents the local isometric embedding by $$x\to(x,z(x)) (x\in R^ n$$, $$z\in R^{n(n-1)/2})$$ then the Gauss equations are $R_{ijkl}(x)=\sum_{\mu}(\frac{\partial^ 2z^ u}{\partial x^ i\partial x^{\mu}}\cdot \frac{\partial^ 2z^{\mu}}{\partial x^ j\partial x^ l}-\frac{\partial^ 2z^{\mu}}{\partial x^ j\partial x^ k}\cdot \frac{\partial^ 2z^{\mu}}{\partial x^ i\partial x^ l})$ which, for $$n=2$$, is a Monge-Ampère equation, and for $$n=3$$ gives 6 equations in the unknowns $$z^ 1,z^ 2,z^ 3$$. The ”deprolonging” of the system $$(I,\omega)$$ is also studied, with special attention to the case $$n=3$$, when $$\Xi_ x$$ are cubic curves in $$P^ 2$$ and the linearized isometric embedding is strictly hyperbolic or of real principal type. Finally, the main result is the following: If $$(M,ds^ 2)$$ is a three dimensional $$C^{\infty}$$ Riemannian manifold and $$x_ 0\in M$$ a point, where the Einstein tensor is not $$(L^ 2)$$, $$(L\in T^ t_ x(M))$$ then there exists a local isometric $$C^{\infty}$$ embedding of a neighborhood of $$x_ 0$$ into $$E^ 6$$.
Reviewer: A.Haimovici

### MSC:

 53B25 Local submanifolds 53B20 Local Riemannian geometry 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces
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### References:

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