Riemannian submanifolds. (English) Zbl 0968.53002

Dillen, Franki J. E. (ed.) et al., Handbook of differential geometry. Volume I. Amsterdam: North-Holland. 187-418 (2000).
This paper is organized as an exposition of results on submanifold theory. It looks exhaustive and covers almost all subjects of the theory and may serve as a good source of references. The material is organized in 22 chapters which are
1. Introduction.
2. Nash’s embedding theorem and some related results.
3. Fundamental theorems, basic notions and results.
4. Rigidity and reduction theorems.
5. Minimal submanifolds.
6. Submanifolds of finite type.
7. Isometric immersions between real space forms.
8. Parallel submanifolds.
9. Standard immersions and submanifolds with simple geodesics.
10. Hypersurfaces of real space forms.
11. Totally geodesic submanifolds.
12. Totally umbilical submanifolds.
13. Conformally flat submanifolds.
14. Submanifolds with parallel mean curvature vector.
15. Kähler submanifolds of Kähler manifolds.
16. Totally real and Lagrangian submanifolds of Kähler manifolds.
17. CR-submanifolds of Kähler manifolds.
18. Slant submanifolds of Kähler manifolds.
19. Submanifolds of the nearly Kähler 6-sphere.
20. Axioms of submanifolds.
21. Total absolute curvature.
22. Total mean curvature.
The author mostly discusses “proper” geometric problems otherwise, for instance, he would pay more attention to the theory of convex surfaces which is known for many deep results and strongly influenced the development of the theory of partial differential equations.
For the entire collection see [Zbl 1069.00010].


53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53B25 Local submanifolds
53C40 Global submanifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds