  We study submanifolds whose principal curvatures, counted with
multiplicities, do not depend on the normal direction. Such submanifolds, which
we briefly call CPC submanifolds, are always austere, hence minimal, and have
constant principal curvatures. Well-known classes of examples include totally
geodesic submanifolds, homogeneous austere hypersurfaces, and singular orbits
of cohomogeneity one actions. The main purpose of this article is to present a
systematic approach to the construction and classification of homogeneous
submanifolds whose principal curvatures are independent of the normal direction
in irreducible Riemannian symmetric spaces of non-compact type and rank greater
than or equal to two. In particular, we provide a large number of new examples
of non-totally geodesic CPC submanifolds not coming from cohomogeneity one
actions (note that only one example was known previously, namely a particular
11-dimensional submanifold of the Cayley hyperbolic plane).
