Killing vector fields — 2 articles found.

On the volume of unit vector fields on Riemannian three-manifolds

C. R. Math. Rep. Acad. Sci. Canada Vol. 30 (1) 2008, pp. 11–21
Vol.30 (1) 2008
Domenico Perrone Details
(Received: 2007-03-08 , Revised: 2008-01-22 )
(Received: 2007-03-08 , Revised: 2008-01-22 )

Domenico Perrone, Dipartimento di Matematica “E. De Giorgi”, Universita del Salento, 73100 Lecce, Italy; email: domenico.perrone@unile.it

Abstract/Résumé:

H. Gluck and W. Ziller proved that the Hopf vector fields, namely, the unit Killing vector fields, are the unique unit vector fields on the unit sphere \(S^3\) that minimize the functional volume. The authors proved this important and famous result by using the method of “calibrated geometries” of Federer and Harvey–Lawson. In this paper, by using a different method, we get an analogue of Gluck and Ziller’s theorem for a compact Sasakian three-manifold with Webster scalar curvature \(w\geq 1\). Moreover, our method gives a new proof of Gluck and Ziller’s theorem. We also extend a theorem of F. Brito about the energy of unit vector fields.

H. Gluck et W. Ziller prouvèrent que les champs de Hopf, c’est-á-dire, les champs vectoriels unitaires de Killing, sont les seuls champs vectoriels unitaires sur la sphére unitaire \(S^3\) que minimisent le volume fonctionnel. Ils prouvèrent cet résultat important en utilisant la méthode des “géométries calibrées” de Federer et Harvey–Lawson. Dans cet article, en utilisant une méthode différente, nous obtenons l’analogue du théorème de Gluck et Ziller pour une 3-variété compact de Sasaki avec courbure scalaire de Webster \(w\geq 1\). En outre, notre méthode donne une nouvelle démonstration du théorème de Gluck et Ziller. Nous aussi étendons un théorème de Brito concernant l’énergie de champs vectoriels unitaires.

Keywords: Killing vector fields, compact Sasakian three-manifolds, energy functional, volume functional

AMS Subject Classification: Global Riemannian geometry; including pinching 53C20

PDF(click to download): On the volume of unit vector fields on Riemannian three-manifolds

Characterization of complex space forms in terms of characteristic vector fields on geodesic spheres

C. R. Math. Rep. Acad. Sci. Canada Vol. 28 (4) 2006, pp. 114–120
Vol.28 (4) 2006
Sadahiro Maeda Details
(Received: 2006-05-30 )
(Received: 2006-05-30 )

Sadahiro Maeda, Department of Mathematics, Shimane University, Matsue 690-8504, Japan; email: smaeda@riko.shimane-u.ac.jp

Abstract/Résumé:

Investigating geometric properties of characteristic vector fields on geodesic spheres in a complex space form, we characterize complex space forms in the class of Kähler manifolds.

En étudiant des propriétés géométriques de champs de vecteurs caractéristiques sur des sphères géodésiques dans un espace complexe à courbure constante, on caractérise ces espaces dans la classe des variétés kähleriennes.

Keywords: Geodesic spheres, Kahler manifolds, Killing vector fields, characteristic fields, complex space forms, totally geodesic complex curves

AMS Subject Classification: Local submanifolds 53B25

PDF(click to download): Characterization of complex space forms in terms of characteristic vector fields on geodesic spheres