Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex Euclidean space

We treat m-dimensional real submanifolds M of complex space forms ̿M when the maximal holomorphic tangent subspace is (m−1)-dimensional. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient space and in this paper we study the condition h(FX,Y)−h(X,FY) = g(FX,Y)η, η∊ T⊥(M), on the structure F and on the second fundamental form h of these submanifolds. Especially when the ambient space ̿M is a complex Euclidean space, we obtain a complete classification of submanifolds M which satisfy these conditions.Mathematics Subject Classifications (2000): 53C15, 53C40, 53B20.     

On a Parametrization Formula for the Solution Set of a Completely Indeterminate Generalized Matricial Carathéodory–Fejér Problem

A parametrization formula for the solution set of a completely indeterminate generalized matricial Carathéodory–Fejér problem is given. The unique A–normalized γ–generating quadruple is constructed by a limit procedure. Moreover, its blocks are expressed as Gramians.