On the energy of unit vector fields with isolated singularities

• Autorzy: Fabiano G. B. Brito , Paweł G. Walczak
• Języki publikacji: EN

Abstrakty

EN

We consider the energy of a unit vector field defined on a compact Riemannian manifold M except at finitely many points. We obtain an estimate of the energy from below which appears to be sharp when M is a sphere of dimension >3. In this case, the minimum of energy is attained if and only if the vector field is totally geodesic with two singularities situated at two antipodal points (at the 'south and north pole').

Słowa kluczowe

EN

Ricci curvature; vector field; mean curvature; energy

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2000
• Tom: 73
• Numer: 3
• Strony: 269-274

Daty

wydano

2000

otrzymano

1999-11-17

Twórcy

autor

Fabiano G. B. Brito

• Departamento de Matemática-IME-USP, Caixa Postal 66281-CEP 05315-970, São Paulo-SP, Brazil

autor

Paweł G. Walczak

• Department of Mathematics, Łódź University, Banacha 22, 90-238 Łódź, Poland

Bibliografia

• [1] F. G. B. Brito, Total bending of flows with mean curvature correction, Differential Geom. Appl. 12 (2000), 157-163.
• [2] J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68.
• [3] P. G. Walczak, An integral formula for a Riemannian manifold with two orthogonal complementary distributions, Colloq. Math. 58 (1990), 243-252.
• [4] G. Wiegmink, Total bending of vector fields on Riemannian manifolds, Math. Ann. 303 (1995), 325-344.
• [5] G. Wiegmink, Total bending of vector fields on the sphere $S^3$, Differential Geom. Appl. 6 (1996), 219-236.
• [6] C. M. Wood, On the energy of a unit vector field, Geom. Dedicata 64 (1997), 319-330.