Total bending of vector fields on Riemannian manifolds. (English) Zbl 0834.53034

Wiegmink, Gerrit

Math. Ann. 303, No. 2, 325-344 (1995).

The functional of total bending of globally defined unit vector fields on a Riemannian manifold is devised to measure to what extent they fail to be parallel. The total bending of such a vector field \(X\) is essentially defined as the integral of the squared norm of the covariant differential of \(X\). Many manifolds do not allow parallel vector fields. Aiming at finding vector fields of minimal total bending, some general variational results are derived and the situation on two-dimensional tori is investigated. In this setting rather complete results are obtained.

Reviewer: G.Wiegmink (Köln)

MSC:

53C20	Global Riemannian geometry, including pinching
53A30	Conformal differential geometry (MSC2010)
58E30	Variational principles in infinite-dimensional spaces

Keywords:

total bending; unit vector fields; two-dimensional tori

References:

[1]	Thierry Aubin, Nonlinear Analysis on Manifolds. Monge-Amp?re Equations. Springer, Grundlehren der mathematischen Wissenschaften 232, New York etc. 1982 · Zbl 0512.53044
[2]	David E. Blair, Contact Manifolds in Riemannian Geometry, Springer Lecture Notes in Mathematics 509, Berlin, Heidelberg 1976 · Zbl 0319.53026
[3]	Shiing-Shen Chern, The Geometry of G-structures, Bull. Am. Math. Soc.72 (1966), 167-219 · Zbl 0136.17804 · doi:10.1090/S0002-9904-1966-11473-8
[4]	James Eells and Luc Lemaire, A Report on Harmonic Maps, Bull. London Math. Soc.10 (1978), 1-68 · Zbl 0401.58003 · doi:10.1112/blms/10.1.1
[5]	James Eells and Luc Lemaire, Another Report on Harmonic Maps, Bull. London Math. Soc.20 (1988), 385-524 · Zbl 0669.58009 · doi:10.1112/blms/20.5.385
[6]	Herman Gluck and Wolfgang Ziller, On the volume of a unit vector field on the three sphere, Comment. Math. Helvetici61 (1986), 177-192 · Zbl 0605.53022 · doi:10.1007/BF02621910
[7]	Sze-Tsen Hu, Homotopy Theory, Academic Press, New York and London 1959 (Volume 8 in the series Pure and Applied Mathematics)
[8]	T?ru Ishihara, Harmonic Sections of Tangent Bundles, J. Math. Tokushima University13 (1979), 23-27 · Zbl 0427.53019
[9]	David L. Johnson, Volumes of Flows, Proc. Amer. Math. Soc.104 (1988), 923-932 · Zbl 0687.58031 · doi:10.1090/S0002-9939-1988-0964875-4
[10]	Jerry L. Kazdan and Frank W. Warner, Curvature functions for compact 2-manifolds, Ann. of Math.99 (1974), 14-47 · Zbl 0273.53034 · doi:10.2307/1971012
[11]	Odette Nouhaud, Applications harmoniques d’une vari?t? riemannienne dans son fibr? tangent. C.R. Acad. Sci. Paris I284 (1977), 815-818 · Zbl 0349.53015
[12]	Sharon L. Pedersen, Volumes of Vector Fields on Spheres, Trans. Amer. Math. Soc.336 (1993), 69-78 · Zbl 0771.53023 · doi:10.2307/2154338
[13]	Walter A. Poor, Differential Geometric Structures, McGraw Hill Book Company, New York etc. 1981 · Zbl 0493.53027
[14]	Norman Steenrod, The Topology of Fibre Bundles, Princeton University Press, Princeton, New Jersey 1951 · Zbl 0054.07103
[15]	Gerrit Wiegmink, Die totale Biegung von Einheitsvektorfeldern und isometrischen Immersionen riemannscher Mannigfaltigkeiten, Dissertation, K?ln 1993
[16]	C. M. Wood, The Gauss Section of a Riemannian Immersion. J. London Math. Soc. (2)33 (1986), 157-168 · Zbl 0607.53036 · doi:10.1112/jlms/s2-33.1.157
[17]	Joseph A. Wolf, Spaces of Constant Curvature, McGraw Hill Book Company, New York etc. 1967 · Zbl 0162.53304
[18]	Gerrit Wiegmink, Total bending of vector fields on the sphere S3, to appear in Differential Geometry and its Applications · Zbl 0866.53025