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On the geometry of affine immersions. (English) Zbl 0629.53012
The authors consider a new approach to affine differential geometry, which extends the notion of relative geometry. An affine immersion $$f: (M^ n,\nabla)\to (\tilde M^{n+1},{\tilde \nabla})$$ is one for which there exists locally a transversal vector field $$\xi$$ along f such that $${\tilde \nabla}_{f_*(X)}f_*(Y)=f_*(\nabla_ XY)+h(X,Y)\xi$$, where $$X, Y$$ are vector fields in $$M$$. By considering this, or the simplified equation $${\tilde \nabla}_ XY=\nabla_ XY+h(X,Y)\xi$$, obtained by dropping $$f_*$$, they introduce the (symmetric) bilinear form $$h$$ on $$TM$$. Similarly, it is also possible to introduce a shape operator $$S$$ and a transversal connection form $$\tau$$ by the further equation $${\tilde \nabla}_ X\xi =-S(X)+\tau (X)\xi.$$
Examples of such setting are: 1) Isometrically immersed hypersurfaces. 2) Affine cylinders: hypersurfaces generated by a family of affine ($$n-1$$)-spaces, each through a point of a curve $$\gamma$$ in $$R^{n+1}$$. 3) Graph immersions, 4) Centro-affine hypersurfaces. 5) Conormal immersions. 6) Blaschke immersions, this is the case usually known in the literature as unimodular affine hypersurface theory. Totally geodesic immersions are defined by the equation $$h=0$$, while graph immersions for a flat connection are characterized by $$S=0.$$
Theorem 1. Let $$f: R^ n\to R^{n+1}$$ be an affine immersion. Then $$\Omega =\{x\in R^ n; S_ x\neq 0$$, $$h_ x\neq 0\}$$, if not empty, is the union of parallel hyperplanes. Each connected component $$\Omega_ a$$ of $$\Omega$$ is a strip consisting of parallel hyperplanes and $$f: \Omega_ a\to R^{n+1}$$ is affinely equivalent to a proper affine cylinder immersion.
Theorem 2 (Cartan-Norden). Let $$(M^ n,g)$$ be a pseudo-Riemannian manifold, $$\nabla$$ its Levi-Civita connection and $$f: (M^ n,\nabla)\to R^{n+1}$$ an affine immersion with a transversal field $$\xi$$. If $$f$$ is nondegenerate we have either (i) $$\nabla$$ is flat and $$f$$ is a graph immersion; or (ii) $$\nabla$$ isnot flat and $$R^{n+1}$$ admits a parallel pseudo-Riemannian metric relative to which f is an isometric immersion and $$\xi$$ is perpendicular to $$f(M^ n)$$. Finally, as Theorem 3, they prove the non-existence of affine immersions into $$R^{n+1}$$ of a compact manifold with an equiaffine connection and strictly negative- definite Ricci tensor.
Reviewer: S.Gigena

##### MSC:
 53A15 Affine differential geometry 53B05 Linear and affine connections
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##### References:
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