  We propose a constrained high-index saddle dynamics (CHiSD) method to search
for index-$k$ saddle points of an energy functional subject to equality
constraints. With Riemannian manifold tools, the CHiSD is derived in a minimax
framework, and its linear stability at an index-$k$ saddle point is proved. To
ensure the manifold property, the CHiSD is numerically implemented using
retractions and vector transport. Then we present a numerical approach by
combining CHiSD with downward and upward search algorithms to construct the
solution landscape in the presence of equality constraints. We apply the
Thomson problem and the Bose-Einstein condensation as numerical examples to
demonstrate the efficiency of the proposed method.
