  We introduce a new technique to the study and identification of submanifolds
of simply-connected symmetric spaces of compact type based upon an approach
computing $k$-positive Ricci curvature of the ambient manifolds and using this
information in order to determine how highly connected the embeddings are. This
provides codimension ranges in which the Cartan type of submanifolds satisfying
certain conditions which generalize being totally geodesic necessarily equals
the one of the ambient manifold. Using results by Guijarro--Wilhelm our
approach partly generalizes recent work by Berndt--Olmos on the index
conjecture.
