On the volume of unit vector fields on spaces of constant sectional
Recommend Documents
May 28, 2015 - DG] 28 May 2015. KILLING VECTOR FIELDS OF CONSTANT LENGTH ON. COMPACT HOMOGENEOUS RIEMANNIAN MANIFOLDS. YU.
folds which carry the space of closed conformal vector fields of at .... âXV = ÏX for all vector fields X. Here â denotes the Levi-Civita connection on. Mn. Then for ...
Oct 2, 2010 - classical mechanics in terms of concepts from symplectic geometry ... of new geometric tools usually referred to as âmultisymplecticâ or âpoly-.
RPq, and we write RPq/RPP for the quotient space obtained from RPq by
identifying ... In ? 7 we compute the values of our operations in projective spaces.
In ...... tIk (). = (1 +. )k -. 1 and k(ps). = ((1 +. )k. 1)s. This completes the proof. In
3 The construction of a Killing vector field. A spacetime admits a Killing vector field Ka whenever the Killing equation. LK gab = âaKb + âbKa = 0. (3.1) holds.
K. DAVID ELWORTHY and Zm-MiNG MA. (Received May 22, 1996) ...... [20] N. Krikorian: Differentiable structures on function spaces, Trans Amer. Math. Soc. 17.
Jun 5, 2011 - manifold there always exists a direct sum decomposition into ... [2, 0.2.25] Let M,N be differentiable manifolds. A product manifold will be.
Jul 12, 2011 - Bogus law Hajduk and Rafa l Walczak. Abstract. We study ..... closed orbit γ which is homologically trivial and /γ η = 0, then [dη]b = 0. The same ...
Amir Baghban. â and Esmail Abedi. 8th Seminar on Geometry and Topology. Amirkabir University of Technology, December 15-17, 2015. Abstract. An isotropic ...
Aug 2, 2015 - sion of a subspace of a × a matrices of rank r. ... Let A, B be k-vector spaces of dimensions a, b (k algebraically closed, of charac- teristic zero). A sub-vector space M ⊂ L(A, B) is said to be of (constant) rank r if every f ∈ M,f =
Jul 27, 2017 - space embedding of the cartesian product of two circles. It lives in R4, .... If (fx,fy)=1. (i.e. the polynomials fx and fy has no common factors), then. P = Af - Cfy, .... The first author is partially supported by a FEDER-MINECO gran
Unit 10. The Lie Algebra of Vector Fields ... fields, Lie algebra of a Lie group. ...
basis of T M at each p e dom x, formed by the tangent vectors {d (p): 1
E' has the Radon-Nikodym property, every weakly compact operator on C(S, E) can be lifted to a weakly .... Departamento de TeorIa de Funciones. Facultad de ...
elements of if at the identity define a subset T of 7.(0) the Lie ... where the Lie bracket is defined as. [(a, A),(b, B)] ..... Let K = SOx(R) X SOâ(R), and let V = R X R".
associate with f the gradient vector field âf on M. For any subset P â M one may ... The main result of the paper (Theorem 3.13) says that the Cech-Alexander ...
Jun 10, 2009 - operator system except that the order units may not be Archimedean. ..... fγ(e) = 1 and fγ(v) = γ, so that fγ is the desired state. D. Remark 2.18.
Dec 3, 2009 - QA] 3 Dec 2009. THE BRAIDED MONOIDAL .... for all x, y, z â G. A 3-cocycle Ï is called normalized if Ï(x, 1,z) = 1, for all x, z â G. Lemma 2.1.
Mar 28, 2015 - by the Dickson Emeriti Professorship at the University of California, .... In our flag domain cases, Proposition 2.1 extends the structural result.
Apr 5, 2005 - this question [27]â[39] from different points of view. In Ref. [16] we .... E2, the angle between this geodesic and a reference geodesic through O [42]. The range of ..... This orbit has always a vertex closest to the focus given by.
Feb 8, 2005 - the full Dushman-Roebuck scheme in the arsenous-acid ex- cess case 22. We assume a high ionic strength for the reac- tion mixture consistent ...
Jul 2, 2017 - Proof. Recall that the differentials of the linear functions generate the ...... with the quantum dynamical process known as the phase damp-.
gave a test of integrability based on the search of systems such that all its solutions ... several kind of systems by the authors MD93], and applied to measure the splitting of ... the Painlev e test cannot detect quadratic integrable systems if the
On the volume of unit vector fields on spaces of constant sectional
a unit vector field on an odd-sphere is always greater than the volume of the radial ... 3 are unit vector fields tangent to classical Hopf fibrations. Hopf vector fields ...
Comment. Math. Helv. 79 (2004) 300–316 0010-2571/04/020300-17 DOI 10.1007/s00014-004-0802-4
c 2004 Birkh¨ auser Verlag, Basel
Commentarii Mathematici Helvetici
On the volume of unit vector fields on spaces of constant sectional curvature Fabiano B. Brito, Pablo M. Chac´ on and A. M. Naveira Abstract. A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima. In this paper, we obtain a lower bound for the volume in terms of the integrals of the 2i-symmetric functions of the second fundamental form of the orthogonal distribution to the field X. In the spheres S2k+1 , this lower bound is independent of X. Consequently, the volume of a unit vector field on an odd-sphere is always greater than the volume of the radial field. The main theorem on volumes is applied also to hyperbolic compact spaces, giving a non-trivial lower bound of the volume of unit fields. Mathematics Subject Classification (2000). 53C20, 58E35. Keywords. Unit vector fields, Sasaki metric, volume functional, spaces forms.
1. Introduction Let M n be a closed oriented Riemannian manifold and X a unit vector field defined on M . The volume of X was defined in [GZ] by vol(X) = vol X(M ) , where X : M → T 1 M is seen as a smooth section on the unit tangent bundle T 1 M , endowed with the Sasaki metric, and vol X(M ) is the volume of the submanifold X(M ) ⊂ T 1 M . A metric formula for the volume of a unit vector field X is given by (see [GZ] and [J]) det(Id + (∇X)t (∇X))ν = vol(X) = M
First author was supported by CNPq (Brazil) Grant No. 301207/80 and by Fapesp (Brazil) Proj. Tem. No. 1999/02684-5. Second and third authors are partially supported by DGI (Spain) Grant No. BFM 2001-3548 and CCEGV (Spain) Grant No. GR00-52. During the elaboration of this paper second author was also partially supported by Fapesp (Brazil).
