A cylinder theorem for a (1, 1)-geodesic affine hypersurface with non-flat induced connection
Given an immersion \( f: M \to \mathbb{R}^{n+p} \) where \( M \) is a manifold of dimension \( n \), and a transversal subbbundle \( N \) of \( T\mathbb{R}^{n+p} \) along \( f \), one can start an affine theory of the pair \( (f,N) \) by decomposing the second derivative of \( f \) in a tangential part (defining the induced torsion-free connection \( \nabla \) on \( M \)) and in a transversal part (defining the vector valued symmetric second fundamental form \( \alpha \)). If it happens that a property is independent of the choice of \( N \) then this property is a genuine affine notion of \( f \) alone. For example, the relative nullity distribution \( \Delta(x) := \{ X \in T_{x}M \mid \alpha(X,Y) = 0 \text{ for all } Y \in T_{x}M \} \) is of this type, and also its dimension \( \nu(x) \), the index of relative nullity. The author studies the case of a complex manifold \( M \) of real dimension \( 2m \) with complex structure \( J \) such that \( \nabla_{X}J = J\nabla_{X} \) and \( \alpha(JX,Y) = \alpha(X,JY) \), calling \( f \) then \( (1,1) \)-geodesic. Also, this notion is independent of \( N \). The main result on these immersions is: If the first normal space has dimension \( 1 \) and a certain regularity condition on the second fundamental form is fulfilled then the nullity index is \( \nu(x) = 2m-2 \). In the hypersurface case (\( p = 1 \)), the authoress then deduces from her former paper [Int. J. Pure Appl. Math. TB{11}, No. 2, 181--204 (2004; Zbl 1057.53010)] that the immersion is \( (2m-2) \)-cylindrical if \( M \) is in addition complete w.r.t. \( \nabla \). The regularity condition is satisfied if \( \nabla \) is non-flat.
scientific article; zbMATH DE number 5675135
scientific article