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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

53A15 Affine differential geometry
53B05 Linear and affine connections
Full Text: DOI EuDML
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