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


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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