Segre embedding and related maps and immersions in differential geometry

Abstract

Segre embedding was introduced by C. Segre in his famous 1891 article [50]. The Segre embedding plays an important roles in algebraic geometry as well as in differential geometry, mathematical physics, and coding theory. In this article, we survey main results on Segre embedding in differential geometry. Moreover, we also present recent differential geometric results on maps and immersions which are constructed in ways similar to Segre embedding. Table of Contents. 1 1. Introduction. 2. Basic formulas and definitions. 3. Differential geometric characterizations of Segre embedding. 4. Degree of Kählerian immersions and homogeneous Kählerian submanifolds via Segre embedding. 5. CR-products and Segre embedding. 6. CR-warped products and partial Segre CR-immersions. 7. Real hypersurfaces as partial Segre embeddings. 8. Complex extensors, Lagrangian submanifolds and Segre embedding. 9. Partial Segre CR-immersions in complex projective space. 10. Convolution of Riemannian manifolds.