On the geometry of affine immersions.

*(English)*Zbl 0629.53012The 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.

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

##### Keywords:

equiaffine structure; shape operator; transversal connection form; totally geodesic immersions; affine immersion; affine cylinder; graph immersion##### References:

[1] | Cartan, E.: Sur la connexion affine des surfaces. C.R. Acad. Sci. Paris178, 242-245 (1924) · JFM 50.0478.01 |

[2] | Kobayashi, S., Nomizu, K.: Foundations of differential geometry, Vol. II. New York: John Wiley 1969 · Zbl 0175.48504 |

[3] | Kulkarni, R.S.: On the principle of uniformization. J. Differ. Geom.13, 109-138 (1978) · Zbl 0381.53023 |

[4] | Nomizu, K.: On completeness in affine differential geometry. Geom. Dedicata20, 43-49 (1986) · Zbl 0587.53010 · doi:10.1007/BF00149271 |

[5] | Nomizu, K., Pinkall, U.: On a certain class of homogeneous projectively flat manifolds, SFB/MPI Preprint 85/44; to appear in T?hoku Math. J. · Zbl 0641.53053 |

[6] | Schirokow, P.A., Schirokow, A.P.: Affine Differentialgeometrie Leipzig: Teubner 1962 |

[7] | Schneider, R.: Zur affinen Differentialgeometrie im Gro?en I. Math. Z.101, 375-406 (1967) · Zbl 0156.20101 · doi:10.1007/BF01109803 |

[8] | Simon, U.: Zur Entwicklung der affinen Differentialgeometrie nach Blachke. Wilhelm Blaschke, Gesammelte Werke, Band 4. Essen: Thales Verlag 1985 |

[9] | Spivak, M.: A comprehensive introduction, to differential geometry, Vol. 5.Publish or Perish Inc., 1975 · Zbl 0306.53001 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.