He was considered an iconic figure in the theory of linear algebraic groups and symmetric spaces.

[2] Although they are often attributed to William Thurston, Satake was the first to introduce orbifold, which he did in the 1950s under the name of V-manifold.

In Satake (1956), he gave the modern definition, along with the basic calculus of smooth functions and differential forms.

He demonstrated that the de Rham theorem and Poincaré duality, along with their proofs, carry over to the orbifold setting.

In Satake (1957), he demonstrated that the standard tensor calculus of bundles, connections, and curvature also carries over to orbifolds, along with the Chern-Gauss-Bonnet theorem and Shiing-Shen Chern's proof thereof.