Submanifolds with restrictions on extrinsic <Emphasis Type="Italic">q</Emphasis>th scalar curvature

We study the structure of the minimum set of the normal curvature for a symmetric bilinear map on Euclidean or Hilbert space, the conditions when this set contains strongly umbilical, conformal nullity, etc. linear subspaces. The main goals are estimates from above of the codimension of these subspaces for a symmetric bilinear map with positive normal curvature and the inequality type restriction on the extrinsic qth scalar curvature. We estimate from above the codimension of asymptotic and relative nullity subspaces for a symmetric bilinear map with nonpositive extrinsic qth scalar curvature. Applying the algebraic results to the second fundamental form of a submanifold with low codimension, we characterize the totally umbilical and totally geodesic submanifolds, prove local nonembedding theorems for the products of Riemannian manifolds and global extremal theorem for the space of positive curvature. On the way we generalize results by Florit (1994), Borisenko (1977, 1987) and Okrut (1991) about Riemannian and Hilbert submanifolds. The research was supported by the Center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel; and by Center for Computational Mathematics and Scientific Computation, University of Haifa.