[2] A Professor Emeritus of Mathematics at University of California, San Diego, Frankel was a longtime member of the Institute for Advanced Study in Princeton, New Jersey.
As a consequence, he was able to prove that, when given a positively curved Riemannian metric on a closed manifold, any two totally geodesic compact submanifolds must intersect if their dimensions are large enough.
By the same approach, Frankel proved that complex submanifolds of positively curved Kähler manifolds must intersect if their dimensions are sufficiently large.
These results were later extended by Samuel Goldberg and Shoshichi Kobayashi to allow positivity of the holomorphic bisectional curvature.
[3] Inspired by work of René Thom, Frankel and Aldo Andreotti gave a new proof of the Lefschetz hyperplane theorem using Morse theory.