# Characterizations of complex space forms by means of geodesic spheres and tubes

Colloquium Mathematicae (1996)

- Volume: 71, Issue: 2, page 253-262
- ISSN: 0010-1354

## Access Full Article

top## Abstract

top## How to cite

topGillard, J.. "Characterizations of complex space forms by means of geodesic spheres and tubes." Colloquium Mathematicae 71.2 (1996): 253-262. <http://eudml.org/doc/210439>.

@article{Gillard1996,

abstract = {We prove that a connected complex space form ($M^n$,g,J) with n ≥ 4 can be characterized by the Ricci-semi-symmetry condition $\tilde\{R\}_\{XY\}·\tilde\{ϱ\}=0$ and by the semi-parallel condition $\tilde\{R\}_\{XY\}·σ=0$, considering special choices of tangent vectors $X,Y$ to small geodesic spheres or geodesic tubes (that is, tubes about geodesics), where $\tilde\{R\}$, $\tilde\{ϱ\}$ and $σ$ denote the Riemann curvature tensor, the corresponding Ricci tensor of type (0,2) and the second fundamental form of the spheres or tubes and where $\tilde\{R\}_\{XY\}$ acts as a derivation.},

author = {Gillard, J.},

journal = {Colloquium Mathematicae},

keywords = {complex space forms; geodesic spheres; tubular hypersurfaces about geodesics},

language = {eng},

number = {2},

pages = {253-262},

title = {Characterizations of complex space forms by means of geodesic spheres and tubes},

url = {http://eudml.org/doc/210439},

volume = {71},

year = {1996},

}

TY - JOUR

AU - Gillard, J.

TI - Characterizations of complex space forms by means of geodesic spheres and tubes

JO - Colloquium Mathematicae

PY - 1996

VL - 71

IS - 2

SP - 253

EP - 262

AB - We prove that a connected complex space form ($M^n$,g,J) with n ≥ 4 can be characterized by the Ricci-semi-symmetry condition $\tilde{R}_{XY}·\tilde{ϱ}=0$ and by the semi-parallel condition $\tilde{R}_{XY}·σ=0$, considering special choices of tangent vectors $X,Y$ to small geodesic spheres or geodesic tubes (that is, tubes about geodesics), where $\tilde{R}$, $\tilde{ϱ}$ and $σ$ denote the Riemann curvature tensor, the corresponding Ricci tensor of type (0,2) and the second fundamental form of the spheres or tubes and where $\tilde{R}_{XY}$ acts as a derivation.

LA - eng

KW - complex space forms; geodesic spheres; tubular hypersurfaces about geodesics

UR - http://eudml.org/doc/210439

ER -

## References

top- [1] E. Boeckx, J. Gillard and L. Vanhecke, Semi-symmetric and semi-parallel geodesic spheres and tubes, Indian J. Pure Appl. Math., to appear. Zbl0866.53037
- [2] B. Y. Chen and L. Vanhecke, Differential geometry of geodesic spheres, J. Reine Angew. Math. 325 (1981), 28-67.
- [3] M. Djorić and L. Vanhecke, Almost Hermitian geometry, geodesic spheres and symmetries, Math. J. Okayama Univ. 32 (1990), 187-206. Zbl0735.53049
- [4] L. Gheysens and L. Vanhecke, Total scalar curvature of tubes about curves, Math. Nachr. 103 (1981), 177-197. Zbl0505.53017
- [5] A. Gray, Tubes, Addison-Wesley, Reading, 1989.
- [6] A. Gray and L. Vanhecke, Riemannian geometry as determined by the volumes of small geodesic balls, Acta Math. 142 (1979), 157-198. Zbl0428.53017
- [7] A. Gray and L. Vanhecke, The volumes of tubes about curves in a Riemannian manifold, Proc. London Math. Soc. 44 (1982), 215-243. Zbl0491.53035
- [8] S. Tanno, Constancy of holomorphic sectional curvature in almost Hermitian manifolds, Kōdai Math. Sem. Rep. 25 (1973), 190-201. Zbl0263.53019
- [9] L. Vanhecke, Geometry in normal and tubular neighborhoods, Rend. Sem. Fac. Sci. Univ. Cagliari, Supplemento al Vol. 58 (1988), 73-176.
- [10] L. Vanhecke and T. J. Willmore, Interaction of tubes and spheres, Math. Ann. 263 (1983), 31-42. Zbl0491.53034
- [11] K. Yano and M. Kon, Structures on Manifolds, Ser. in Pure Math. 3, World Sci., Singapore, 1984.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.