de la SOCIÉTÉ MATHÉMATIQUE DE FRANCE UNIT FIELDS ON PUNCTURED SPHERES Fabiano G.B. Brito & Pablo M. Chacón & David L. Johnson

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1 Bulletin de la SOCIÉTÉ MATHÉMATIQUE DE FRANCE UNIT FIELDS ON PUNCTURED SPHERES Fabiano G.B. Brito & Pablo M. Chacón & David L. Johnson Tome 36 Fascicule 8 SOCIÉTÉ MATHÉMATIQUE DE FRANCE Publié avec le concours du Centre national de la recherche scientifique pages 47-57

2 Bull. Soc. math. France 36 (), 8, p UNIT VECTOR FIELDS ON ANTIPODALLY PUNCTURED SPHERES: BIG INDEX, BIG VOLUME by Fabiano G.B. Brito, Pablo M. Chacón & David L. Johnson Abstract. We establish in this paper a lower bound for the volume of a unit vector field v defined on S n \ {±x}, n =, 3. This lower bound is related to the sum of the absolute values of the indices of v at x and x. Résumé (Champs unitaires dans les sphères antipodalement trouées : grand indice entraîne grand volume) Nous établissons une borne inférieure pour le volume d un champ de vecteurs v défini dans S n \ {±x}, n =, 3. Cette borne inférieure dépend de la somme des valeurs absolues des indices de v en x et en x. Texte reçu le 9 septembre 6, révisé le avril 7 Fabiano G.B. Brito, Dpto. de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, R. do Matão, São Paulo-SP, (Brazil) fabiano@ime.usp.br Pablo M. Chacón, Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced -4, 378 Salamanca (Spain) pmchacon@usal.es David L. Johnson, Department of Mathematics, Lehigh University, 4 E. Packer Avenue, Bethlehem, PA, 85 (USA) david.johnson@lehigh.edu Mathematics Subject Classification. 53C, 57R5, 53C. Key words and phrases. Unit vector fields, volume, singularities, index. During the preparation of this paper the first author was supported by CNPq, Brazil. The second author is partially supported by MEC/FEDER project MTM C4-, Spain. The third author was supported during this research by grants from the Universidade de São Paulo, FAPESP Proc. 999/684-5, and Lehigh University, and thanks those institutions for enabling the collaboration involved in this work. BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE /8/47/$ 5. Société Mathématique de France

3 48 BRITO (F.G.B.), CHACÓN (P.M.) & JOHNSON (D.L.). Introduction The volume of a unit vector field v on a closed Riemannian manifold M is defined [] as the volume of the section v : M T M, where the Sasakian metric is considered in T M. The volume of v can be computed from the Levi-Civita connection of M. If we denote by ν the volume form, for an orthonormal local frame {e a } n a=, we have ( n () vol( v ) = + ea v + M a= + + a < <a n a <a ea v ea v ea v ean v ) ν. Note that vol( v ) vol(m) and also that only parallel fields attain the trivial minimum. For odd-dimensional spheres, vector fields homologous to the Hopf fibration v H have been studied, see [], [3], [9] and []. In [5], a non-trivial lower bound of the volume of unit vector fields on spaces of constant curvature was obtained. In S k+, only the vector field n tangent to the geodesics from a fixed point (with two singularities) attains the volume of that bound. We call this field n north-south or radial vector field. We notice that unit vector fields with singularities show up in a natural way, see also []. For manifolds of dimension 5, a theorem showing how the topology of a vector field influences its volume appears in [4]. More precisely, the result in [4] is an inequality relating the volume of v and the Euler form of the orthogonal distribution to v. The purpose of this paper is to establish a relationship between the volume of unit vector fields and the indices of those fields around isolated singularities. We consider these notes to be a preliminary effort to understand this phenomenon. For this reason, we have chosen a simple model where such a relationship is found. We hope this could serve as inspiration for more complex situations to be treated in a near future. Precisely, we prove here: Theorem.. Let W = S n \{N, S}, n = or 3, be the standard Euclidean sphere where two antipodal points N and S are removed. Let v be a unit smooth vector field defined on W. Then, for n =, vol( v ) ( π + I v (N) + I v (S) ) vol(s ); for n = 3, vol( v ) ( I v (N) + I v (S) ) vol(s 3 ), where I v (P ) stands the Poincaré index of v around P. tome 36 8 n o

4 UNIT FIELDS ON PUNCTURED SPHERES 49 It is easy to verify that the north-south field n achieves the equalities in the theorem. In fact, the volume of n in S is equal to πvol(s ), and in S 3 is vol(s 3 ). We have to point out that vol( n) = vol( v H ) in S 3. The lower bound in S 3 when the singularities are trivial (i.e. I v (N) = I v (S) = ) has no special meaning. We will comment briefly some possible extensions for this result in Section 3 of this paper.. Proof of the theorem A key ingredient in the proof of the theorem is the application of the following result of Chern [7]. The second part of this statement is a special case of the result of Section 3 of that article. Proposition. (see Chern [7]). Let M n be an orientable Riemannian manifold of dimension n, with Riemannian connection -form ω and curvature form Ω. Then, there is an (n )-form Π on the unit tangent bundle T M with π : T M M the bundle projection, so that: ß e(ω) if n is even, dπ = if n is odd. In addition, π (x) Π = for any x M, that is, Π π (x) is the induced volume form of the fiber π (x), normalized to have volume. The form Π as described by Chern is somewhat complicated. First, define forms φ k for k {,..., [ n] }, by choosing a frame {e,..., e n } of T M, so that {e,..., e n } frame π (x) at e n π (x). Then, at e n T M, φ k = ɛ α...α n Ω αα Ω αk α k ω αk+ n ω αn n, α,...,α n n where ɛ α...α n is the sign of the permutation, and from this n ( ) k π n 3 (n k ) k+ n k! φ k if n is even, k= Π = (n ) ( ( ) k (n ) ) φ n π (n ) ( k if n is odd. (n ))! k k= Subsequent treatments of this general theory [8], [] use more elegant formulations of forms similar to this, but usually only for the bundle of frames, and avoid the case where M is odd-dimensional. BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE

5 5 BRITO (F.G.B.), CHACÓN (P.M.) & JOHNSON (D.L.) The cases relevant to this research are for n = and n = 3, where these formulas simplify to { π Π = ω if n =, 4π (ω 3 ω 3 Ω ) if n = 3. Even though there is a common line of reasoning in the proof of both parts of the theorem, each dimension has its special features. For that reason, we provide separate proofs for dimensions and 3... Case n =. Denote by g the usual metric on S induced from R 3. Without loss of generality we take N = (,, ) and S = (,, ). On W we consider an oriented orthonormal local frame {e, e = v }. Its dual basis is denoted by {θ, θ } and the connection -forms of are ω ij (X) = g( X e j, e i ) for i, j =, where X is a vector in the corresponding tangent space. In dimension, the volume () reduces to: vol( v ) = + k + τ ν, S where k = g( v v, e ) is the geodesic curvature of the integral curves of v and τ = g( e v, e ) is the geodesic curvature of the curves orthogonal to v. Also, ω = τθ + kθ. The first goal is to relate the integrand of the volume with the connection form ω. If Sϕ is the parallel of S at latitude ϕ ( π, π) consider the unit field u on Sϕ such that { u, n} is positively oriented where n is the field pointing toward N. Let α [, π] be the oriented angle from u to v. Then u = sin αe + cos α v. If i : Sϕ S is the inclusion map, we have () i ω ( u) = τθ ( u) + kθ ( u) = τ sin α + k cos α. We split the domain of the integral in northern and southern hemisphere, H + and H respectively. First we consider the northern hemisphere H +. From the general inequality a + b a cos β + b sin β a cos β + b sin β, for any a, b, β R, we have: (3) + k + τ cos ϕ + k + τ sin ϕ cos ϕ + k cos α + τ sin α sin ϕ = cos ϕ + i ω ( u) sin ϕ. tome 36 8 n o

6 UNIT FIELDS ON PUNCTURED SPHERES 5 Denote by ν the induced volume form to S ϕ. From () and (3) we get (4) ( vol( v ) H + cos ϕ + i ω ( u) sin ϕ ) ν H + π = cos ϕν π dϕ + i ω ( u) sin ϕν dϕ Sϕ π π cos ϕdϕ + π S ϕ sin ϕ S ϕ i ω dϕ. The connection form ω satisfies dω = θ θ. Therefore, the area of the annulus region A ( ϕ, π ɛ) = { (x, x, x 3 ) S sin ϕ x 3 sin( π ɛ)} provides the equality (5) A(ϕ, ) dω = area of A = ϕ π cos tdt = π ( sin( π ɛ) sin ϕ). The boundary of A(ϕ, π ɛ) is A = S ϕ S (with the appropriate orientation), so by (5) and Stokes Theorem (6) S ϕ i ω = A(ϕ, ) dω + S i ω = π ( sin ( π ɛ) sin ϕ ) + S i ω. If ω is the Riemannian connection form of the standard metric on S, since the limit as ɛ goes to of v S maps S onto the fiber I v (N) times, from Proposition. we have lim ɛ i ω = π lim S ɛ i v Π = πi v (N) S π (N) Π = πi v (N). Thus, from (6) (7) S ϕ i ω = π( sin ϕ) + πi v (N). BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE

7 5 BRITO (F.G.B.), CHACÓN (P.M.) & JOHNSON (D.L.) Following from (4) with (7) we have: (8) vol( v ) H + π + = π + π + π + π π π π = π + π I v (N) = π sin ϕ π( sin ϕ) + πi v (N) dϕ π sin ϕi v (N) π sin ϕ(sin ϕ ) dϕ π sin ϕi v (N) π sin ϕ(sin ϕ ) dϕ ( π sin ϕi v (N) π sin ϕ(sin ϕ ) ) dϕ π π + I v (N) π π +. π sin ϕdϕ π (sin ϕ sin ϕ)dϕ For the southern hemisphere, the index of v at S is obtained by i ω = vol(s )I v (S). lim ɛ S π+ɛ Therefore, if π < ϕ we have (9) i ω = πi v (S) π(sin ϕ + ). S ϕ In order to obtain a similar equation to (3) we take β = ϕ, and together with () we have ( () vol( v ) H cos ϕ i ω ( u) sin ϕ ) ν H π cos ϕdϕ i ω sin ϕdϕ. π π From (9) and (): () vol( v ) H π πi v (S) π(sin ϕ + ) sin ϕdϕ π π π + I v (S) sin ϕ dϕ π sin ϕ + sin ϕ dϕ π π = π π + I v (S) π + π. S ϕ tome 36 8 n o

8 UNIT FIELDS ON PUNCTURED SPHERES 53 Finally, recall that the sum of the indices of a field in S must be, therefore the sum of the absolute values of the indices must be greater or equal than. So, from (8) and (), the volume of v is bounded by vol( v ) π + π I v (N) π + π + π I v (S) π + π π + πi v (N) + πi v (S) 4π + π = π + π ( I v (N) + I v (S) ) + π = π + π ( I v (N) + I v (S) ) = ( π + I v (N) + I v (S) ) vol(s ).. Case n = 3. As before, denote by g the metric in S 3 and consider a general situation where N = (,,, ), S = (,,, ) and I v (N) (and therefore I v (S) ). If v is a unit vector field on W, consider on W an oriented orthonormal local frame such that {e, e, e 3 = v}. The dual basis will be denoted by {θ, θ, θ 3 }. The coefficients of the second fundamental form of the orthogonal distribution to v, possibly non-integrable, are h ij = ω i3 (e j ) = g( ej v, e i ). The coefficients of the acceleration of v are given by v v = a e +a e. Finishing the notation, we will use J for the integrand of the volume () and h h σ = h h, σ h a, = h a, σ a h, = a h. It is easy to see that ( J = + i,j= h ij + a + a + σ + (σ, ) + (σ, ) ). Note that ( + σ ) = + σ + σ + () J h ij + σ. Therefore i,j=» ( + σ ) + σ,, where equality holds if and only if a = a = and we have either h = h and h = h, or h = h and h = h. Now we want to identify the last term in () with the evaluation of certain forms. In the frame {e, e, v} we can demand that e will be tangent to S ϕ, the parallel of S 3 with latitude ϕ ( π, π). We complete a frame in S ϕ with u in such a way {e, u} is an oriented local frame compatible with the normal field n that points toward the North Pole. That is, in such a way that {e, u, n} BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE

9 54 BRITO (F.G.B.), CHACÓN (P.M.) & JOHNSON (D.L.) is a positively oriented local frame of S 3. Let α [, π] be the oriented angle from T Sϕ to v and i : Sϕ S 3 the inclusion map. In this way, u = cos α v + sin αe and i (θ θ )(e, u) = sin α, i (θ θ 3 )(e, u) = cos α, i (θ θ 3 )(e, u) =. In order to evaluate i (ω 3 ω 3 ), first we note that ω 3 ω 3 = σ θ θ + σ, θ θ 3 σ, θ θ 3. So, i (ω 3 ω 3 )(e, u) = sin ασ + cos ασ,. As in (3) with β [, π] such that sin β = sin α and cos β = cos α, from () we get (3) J sin β ( + σ ) + cos β σ, = sin α + sin α σ + cos α σ, sin α + sin ασ + cos ασ, = i (θ θ )(e, u) + i (ω 3 ω 3 )(e, u) ( i (θ θ ) + i (ω 3 ω 3 ) ) (e, u). We split W in northern and southern hemisphere, H + and H respectively. Then, from (3) ( (4) vol( v ) H + i (θ θ ) + i (ω 3 ω 3 ) ) (e, u) ν H + π ( i (θ θ ) + i (ω 3 ω 3 ) ) dϕ. S ϕ We know that dω = ω 3 ω 3 + θ θ. If A(ϕ, π ɛ) is the annulus region between the parallels Sϕ and S, ϕ < π ɛ < π, we have by Stokes Theorem (5) i (ω 3 ω 3 ) + i (θ θ ) = i (ω 3 ω 3 ) + i (θ θ ). S ϕ S S We bound i (θ θ )(e, u) = sin α on S and consequently i (ω 3 ω 3 ) + i (θ θ ) i (ω 3 ω 3 ) 4π cos ( π ɛ). S ϕ S tome 36 8 n o

10 UNIT FIELDS ON PUNCTURED SPHERES 55 Applying Proposition., since as before, the limit as ɛ goes to of v S maps S onto the fiber I v(n) times and noting that the curvature term is horizontal so goes to in the limit, i (ω 3 ω 3 ) = lim i (ω 3 ω 3 Ω ) lim ɛ So, (6) S S ϕ From (4) and (6) we get (7) vol( v ) H + ɛ S = 4π lim i v Π = 4πI v (N) ɛ S π (N) i (ω 3 ω 3 ) + i (θ θ ) 4πI v (N). π 4π I v (N) dϕ = π I v (N). Π = 4πI v (N). In a similar way for the southern hemisphere, the integral of dω over the annulus region A( π + ɛ, ϕ), π < π + ɛ < ϕ provides exactly (5) but now we bound sin α to obtain i (ω 3 ω 3 ) + i (θ θ ) i (ω 3 ω 3 ) + 4π cos ( π + ɛ). S ϕ S π+ɛ The index of v at S can be calculated as i (ω 3 ω 3 ) = vol(s )I v (S). So, Therefore, (8) S ϕ lim ɛ vol( v ) H S π+ɛ i (ω 3 ω 3 ) + i (θ θ ) 4πI v (S). π Thus, from (7) and (8) we have S ϕ i (θ θ ) + i (ω 3 ω 3 ) dϕ π 4π I v (S) dϕ = π I v (S). vol( v ) π ( I v (N) + I v (S) ) = ( I v (N) + I v (S) ) vol(s 3 ). BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE

11 56 BRITO (F.G.B.), CHACÓN (P.M.) & JOHNSON (D.L.) 3. Concluding remarks These results should extend to higher dimensions if one makes use of some rather complicated inequalities involving the volume integrand in () of a unit vector field and some symmetric functions coming from the second fundamental form of the orthogonal distribution (which is generally non integrable). Some of these inequalities can be found in [6] or [5]. Index results should exist also for the case when the spheres are punctured differently. In other words, if we have two singularities which are not antipodal points of S or S 3 or if we have more than two singularities, what could be said? We believe that some results relating indices and positions of the singularities to the volume of a unit vector field may be found. For singular vector fields on S another natural situation is the one of unit vector fields defined on S \ {x}. In a recent paper [], see also [], a unit vector field p is defined on S \ {x} by parallel translation of a given tangent vector at x along the minimizing geodesics to x. It has been proved in [] that p minimizes the volume of unit vector fields defined on S \ {x}. By a direct calculation, we obtain the inequality vol( p) > vol( n), where n is the north-south vector field tangent to the longitudes of W. Now, new questions arise about minimality on specific topological-geometrical configurations on the punctured spheres. BIBLIOGRAPHY [] V. Borrelli & O. Gil-Medrano Area minimizing vector fields on round -spheres, 6, preprint. [], A critical radius for unit Hopf vector fields on spheres, Math. Ann. 334 (6), p [3] F. G. B. Brito Total bending of flows with mean curvature correction, Differential Geom. Appl. (), p [4] F. G. B. Brito & P. M. Chacón A topological minorization for the volume of vector fields on 5-manifolds, Arch. Math. (Basel) 85 (5), p [5] F. G. B. Brito, P. M. Chacón & A. M. Naveira On the volume of unit vector fields on spaces of constant sectional curvature, Comment. Math. Helv. 79 (4), p [6] P. M. Chacón Sobre a energia e energia corrigida de campos unitários e distribuições. Volume de campos unitários, Ph.D. Thesis, Universidade de São Paulo, Brazil,, and Universidad de Valencia, Spain,. tome 36 8 n o

12 UNIT FIELDS ON PUNCTURED SPHERES 57 [7] S. S. Chern On the curvatura integra in a Riemannian manifold, Ann. of Math. () 46 (945), p [8] S. S. Chern & J. Simons Characteristic forms and geometric invariants, Ann. of Math. () 99 (974), p [9] O. Gil-Medrano & E. Llinares-Fuster Second variation of volume and energy of vector fields. Stability of Hopf vector fields, Math. Ann. 3 (), p [] H. Gluck & W. Ziller On the volume of a unit vector field on the three-sphere, Comment. Math. Helv. 6 (986), p [] D. L. Johnson Chern-Simons forms on associated bundles, and boundary terms, Geometria Dedicata (7), p [] S. L. Pedersen Volumes of vector fields on spheres, Trans. Amer. Math. Soc. 336 (993), p BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE

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