[2] His research interests were in Riemannian and complex manifolds, transformation groups of geometric structures, and Lie algebras.

Kobayashi served as chairman of the Berkeley Mathematics Department for a three-year term from 1978 to 1981 and for the 1992 Fall semester.

The two-volume book Foundations of Differential Geometry, which he coauthored with Katsumi Nomizu, has been known for its wide influence.

In this setting, Shiing-Shen Chern, Manfredo do Carmo, and Kobayashi studied the algebraic structure of the zeroth-order terms, showing that they are nonnegative provided that the norm of the second fundamental form is sufficiently small.

[5] On a Kähler manifold, it is natural to consider the restriction of the sectional curvature to the two-dimensional planes which are holomorphic, i.e. which are invariant under the almost-complex structure.

In particular they established, by the Bochner technique, that the second Betti number of a connected closed manifold must equal one if there is a Kähler metric whose holomorphic bisectional curvature is positive.

This, in combination with the Goldberg–Kobayashi result, forms the final part of Yum-Tong Siu and Shing-Tung Yau's proof of the Frankel conjecture.

[6] Kobayashi and Ochiai also characterized the situation of c1(M) = nα as M being biholomorphic to a quadratic hypersurface of complex projective space.