The canonical cone structure on a compact Hermitian symmetric space G/P is the fiber bundle Open image in new window where Open image in new window is the cone of the highest weight vectors under the action of the reductive part of P. It is known that the cone Open image in new window coincides with the cone Open image in new window of the vectors tangent to the lines in G/P passing through x, when we consider G/P as a projective variety under its homogeneous embedding into the projective space Open image in new window of the irreducible representation space Vλ of G with highest weight λ associated to P. A subvariety X of G/P is said to be an integral variety of Open image in new window at all smooth points x∈G/P. Equivalently, an integral variety of Open image in new window is a subvariety of G/P whose embedded projective tangent space at each smooth point is a linear space Open image in new window We prove a kind of rigidity of the integral varieties under some dimension condition. After making a uniform setting to study the problem, we apply the theory of Lie algebra cohomology as a main tool. Finally we show that the dimension condition is necessary by constructing counterexamples.