Die Bedeutung Christoffelscher Zusammenhänge in der affinen Differentialgeometrie

Blaschke’s affine differential geometry is related to an affine space with a volume measure. First we give a summary how this affine geometry can be generalized in a manifold with a volume measure and a volume preserving connection. In this matter a hypersurface turns out to be characterizable by a quadratic form and an induced connection in a more consequent affine geometrical way. (instead of by quadratic and cubic forms). In order to let the consequences of this development become clear also for the classical affine differential geometry, in the second part we present the theory of hypersurfaces in the n-dimensional affine space in detail under these points of view. Thus one is able to follow up a special situation of the different ways in which Christoffel’s ideas are met within affine differential geometry.