Poincaré index and the volume functional of unit vector fields on punctured spheres
PPOINCAR ´E INDEX AND THE VOLUME FUNCTIONAL OF UNITVECTOR FIELDS ON PUNCTURED SPHERES
FABIANO G. B. BRITO, ANDR´E O. GOMES, AND ICARO GONC¸ ALVES
Abstract.
For n ≥
1, we exhibit a lower bound for the volume of a unit vector field on S n +1 \{± p } depending on the absolute values of its Poincar´e indices around ± p . We deter-mine which vector fields achieve this volume, and discuss the idea of having multiple isolatedsingularities of arbitrary configurations. Introduction and statement of the results
Let M m be a closed oriented Riemannian manifold and (cid:126)v a unit vector field on M . If T M denotes the unit tangent bundle, endowed with the Sasaki metric, and regarding (cid:126)v : M → T M as a smooth section, the volume of (cid:126)v is defined as the volume of the submanifold (cid:126)v ( M ) ⊂ T M ,vol( (cid:126)v ) = vol( (cid:126)v ( M )) . On a given orthonormal local frame { e , . . . e m } , there exists a formula (see [8] and [9]) interms of the Riemannian metric of M . It readsvol( (cid:126)v ) = (cid:90) M » det(Id + ( ∇ (cid:126)v ) ∗ ( ∇ (cid:126)v )) ν = (cid:90) M Å (cid:88) A (cid:107)∇ e A (cid:126)v (cid:107) + (cid:88) A
Theorem 2 ([9]) . Hopf fibrations on the round sphere S are not local minima of the volumefunctional. In pursuit of unit vector fields of minimum volume, several constructions stumbled onspheres minus one or minus a couple of points. One must keep in mind the following twoexamples, both of them defined on punctured spheres.The first example was given by Pedersen in [10], defined on a sphere minus one point. Wedenote it by V P . It was shown in [10] that its volume isvol( V P ) = √ πn vol( S n +1 ) , for n ≥
1. The second example is a radial vector field on S n +1 \{± p } . This vector field, denotedby V R , is a geodesic vector field coming from the exponential map of the sphere at p . Brito etal proved the following Theorem 3 ([5]) . Let (cid:126)v be a unit vector field on a compact Riemannian and oriented manifold M n +1 . Then vol( (cid:126)v ) ≥ (cid:90) M Ñ n (cid:88) k =0 (cid:32) nk (cid:33)(cid:32) n k (cid:33) − | σ k ( (cid:126)v ⊥ ) | é ν, where σ k ( (cid:126)v ⊥ ) is the k -th elementary symmetric function of the second fundamental form ofthe distribution orthogonal to (cid:126)v (that is not necessarily integrable), with σ = 1 . When n ≥ , OWER BOUND FOR THE VOLUME OF UNIT VECTOR FIELDS ON PUNCTURED SPHERES 3 equality holds if and only if (cid:126)v is totally geodesic and (cid:126)v ⊥ is integrable and umbilic. Furthermore,the following holds,(a) For every unit vector field (cid:126)v on S n +1 , vol( (cid:126)v ) ≥ n (cid:88) k =0 (cid:32) nk (cid:33) (cid:32) n k (cid:33) − vol( S n +1 ) , and for n ≥ none of them achieves equality.(b) Let (cid:126)v be any non-singular unit vector field on vol( S n +1 ) , then vol( V R ) ≤ vol( (cid:126)v ) . Besides, singular unit vector fields on S and the influence of the radius of a given sphereon the volume of Hopf vector fields have been studied, [1] and [2].It can be shown that vol( V R ) = n ( nn ) vol( S n +1 ) (for example, see [5]). Together with thevalue computed in [8] for Hopf vector fields, vol( V H ) = 2 n vol( S n +1 ), one is able to summarizesome inequalities vol( S n +1 ) < vol( V R ) < vol( V P ) (cid:28) vol( V H ) , whenever n ≥ N = p and S = − p , Theorem 4 ([4]) . Let (cid:126)v be a unit smooth vector field defined on S m \{ N, S } . Then(a) for m = 2 , vol( (cid:126)v ) ≥ ( π + | I (cid:126)v ( N ) | + | I (cid:126)v ( S ) | − S ) ,(b) for m = 3 , vol( (cid:126)v ) ≥ ( | I (cid:126)v ( N ) | + | I (cid:126)v ( S ) | )vol( S ) ,where I (cid:126)v ( P ) stands for the Poincar´e index of (cid:126)v around P . Our main goal is to extend the above result to higher odd dimensional spheres. The maintheorem asserts
FABIANO G. B. BRITO, ANDR´E O. GOMES, AND ICARO GONC¸ ALVES
Theorem A. If (cid:126)v is a unit vector field on S n +1 \{± p } , then (2) vol( (cid:126)v ) ≥ π S n ) ( | I (cid:126)v ( p ) | + | I (cid:126)v ( − p ) | ) . In comparing the above estimate to the value achieved by radial vector fields, the followingconsequence is deduced
Corollary 1.
For any unitary vector field (cid:126)v on S n +1 \{± p } , vol( (cid:126)v ) ≥ vol( V R )2 ( | I (cid:126)v ( p ) | + | I (cid:126)v ( − p ) | ) , where V R denotes the north-south vector field. The technique presented here can be exploited to obtain a straightforward extension toarbitrary isolated singularities, in a general Riemannian compact manifold
Theorem B.
Let (cid:126)v be a unit vector field defined on M n +1 \{∪ mi =1 p i } , where M is a compactRiemannian manifold and { p i } is a subset of isolated points. Then (3) vol( (cid:126)v ) ≥ vol( S n )2 m (cid:88) i =1 | I (cid:126)v ( p i ) | This paper is organized as follows. We start Section 2 by introducing the Euler class ofthe normal bundle of (cid:126)v , and then we define a list of functions depending on the vector field.We finish this Section by exhibiting an explicit representative of the Euler class. Section 3 isdivided in five subsections, and in the last two of them we prove theorems A and B, respectively.Subsection 3.1 is devoted to show how the indices of the vector field arise when the Euler classis restricted to small neighborhoods around its singularities. In Subsection 3.2 we briefly reviewsome results from [5] and use them to establish a comparison between the integrand in 1 and afunction determined by the restriction of the Euler class. Last Section is dedicated to discussthe main theorems and future developments as well.
OWER BOUND FOR THE VOLUME OF UNIT VECTOR FIELDS ON PUNCTURED SPHERES 5 Preliminaries and the Euler class
Let n ≥ M := S n +1 \{± p } , endowed with the Riemannian metric (cid:104)· , ·(cid:105) . Let (cid:126)v be a unit vector field (cid:126)v : M → T M , and take { e , . . . , e n , e n +1 = (cid:126)v } as an orthonormal localframe. We fix the following notation: 1 ≤ i, j, k, l, · · · ≤ n and 1 ≤ A, B, C, D, · · · ≤ n + 1.If { ω A } is the associated local coframe, then the curvature and connection forms are related bythe structure equations of M , ω A ( e B ) = δ AB , δ AB = 0 if A (cid:54) = B, δ AA = 1 , ∇ e A = (cid:88) B ω AB e B , ω AB + ω BA = 0 ,dω A = (cid:88) B ω AB ∧ ω B , dω AB = (cid:88) C ω AC ∧ ω CB − Ω AB , Ω AB = 12 (cid:88) C,D R ABCD ω C ∧ ω D , R ABCD + R ABDC = 0 , The normal bundle (cid:126)v ⊥ is a subbundle of T M , and it admits a natural second fundamentalform given locally by the matrix ( a ij ), constructed with respect to the aforementioned localframe, a AB = (cid:104)∇ e B (cid:126)v, e A (cid:105) . The curvature form of (cid:126)v ⊥ , Ω ⊥ AB , is related to Ω AB by means of thestructure equations,(4) Ω ⊥ AB = Ω AB + ω A n +1 ∧ ω B n +1 . We recall the definition of the Euler form in terms of the Pfaffian of Ω ⊥ AB ,(5) E ( (cid:126)v ⊥ ) = 2(2 n )!vol( S n ) (cid:88) σ ∈S n sgn( σ )Ω ⊥ σ (1) σ (2) ∧ · · · ∧ Ω ⊥ σ (2 n − σ (2 n ) , where S n stands for the permutation group of 2 n elements while sgn( σ ) equals the sign of σ .Before computing E ( (cid:126)v ⊥ ) we need to settle our notation. For each 1 ≤ i ≤ n , we say that σ i is the i -th elementary symmetric function of the matrix ( a ij ). The function σ i is the sum ofall i × i minors from ( a ij ).The last column of ( a AB ) has some special meaning. It is formed by the elements a i n +1 = (cid:104)∇ (cid:126)v (cid:126)v, e i (cid:105) , which are components of the acceleration of (cid:126)v . We employ these components in thenext definition. FABIANO G. B. BRITO, ANDR´E O. GOMES, AND ICARO GONC¸ ALVES
Definition.
Let ( a ij ( l )) denote the n × n matrix obtained from ( a ij ) by changing its l -thcolumn with the components of ∇ (cid:126)v (cid:126)v , ( a ij ( l )) = à a · · · a l − a n +1 a l +1 · · · a n ... ... ... ... ... a n · · · a n l − a n n +1 a n l +1 · · · a n n í . We say that σ ⊥ i ( l ) is the sum of all i × i minors of the matrix ( a ij ( l )) having at least oneelement depending on ∇ (cid:126)v (cid:126)v . For example, σ ⊥ (2 n ) is the sum of all 2 × a ij (2 n ) such that at least one of theircolumns is made of components of ∇ (cid:126)v (cid:126)v , σ ⊥ (2 n ) = n (cid:88) j =11 ≤ i The Euler class E ( (cid:126)v ⊥ ) ∈ H n ( M, R ) = H n ( S n +1 \{± p } , R ) ∼ = R can be representedby the following element (6) E ( (cid:126)v ⊥ ) = 2vol( S n ) n (cid:88) k =0 (cid:32) nk (cid:33)(cid:32) n k (cid:33) − W ( k ) , where, denoting (cid:98) ω the omitted term, W ( k ) = (cid:88) C σ ⊥ k ( C ) ω ∧ · · · ∧ ” ω C ∧ · · · ∧ ω n +1 = (cid:88) l σ ⊥ k ( l ) ω ∧ · · · ∧ (cid:99) ω l ∧ · · · ∧ ω n +1 + σ k ω ∧ · · · ∧ ω n . Proof. The fact that Ω AB = ω A ∧ ω B (the metric on M is just the restriction of the roundRiemannian metric of S n +1 ) together with a nice rearrangement of terms imply E ( (cid:126)v ⊥ ) = 2(2 n )!vol( S n ) (cid:88) σ ∈S n sgn( σ ) n (cid:88) k =0 (cid:32) nk (cid:33) ω σ (1) ∧ · · · ∧ ω σ (2 k ) ∧ ω σ (2 k +1) 2 n +1 ∧ · · · ∧ ω σ (2 n ) 2 n +1 . OWER BOUND FOR THE VOLUME OF UNIT VECTOR FIELDS ON PUNCTURED SPHERES 7 Taking the second fundamental form of (cid:126)v ⊥ into account, we write ω A n +1 = − (cid:80) B a AB ω B ,and consequently ω A n +1 ∧ ω B n +1 = (cid:80) C,D a AC a BD ω C ∧ ω D . Hence E ( (cid:126)v ⊥ ) = 2(2 n )!vol( S n ) (cid:88) σ ∈S n sgn( σ ) n (cid:88) k =0 (cid:32) nk (cid:33) ω σ (1) ∧ · · · ∧ ω σ (2 k ) ∧ Ñ (cid:88) B a σ (2 k +1) B ω B é ∧ · · · ∧ Ñ (cid:88) B n − k ) a σ (2 n ) B n − k ) ω B n − k ) é . Now it is a matter of separating the coefficients of 2 n -forms ω A ∧ · · · ∧ ω A n .When we fix those 2 n -forms, we have to count them within all permutations in S n . Forexample, k = 1 gives us the following summand (cid:88) σ ∈S n sgn( σ ) ω σ (1) ∧ ω σ (2) ∧ Ñ (cid:88) B a σ (3) B ω B é ∧ · · · ∧ Ñ (cid:88) B n − a σ (2 n ) B n − ω B n − é . Consequently, we end up with a number, (2 n − k )!(2 k )!, and since the Pfaffian is dividedby (2 n )! we have that (2 n − k )!(2 k )!(2 n )! = Ä n k ä − .On the other hand, the products a σ (2 k +1) B · · · a σ (2 n ) B n − k ) from Ñ (cid:88) B a σ (2 k +1) B ω B é ∧ · · · ∧ Ñ (cid:88) B n − k ) a σ (2 n ) B n − k ) ω B n − k ) é determine some minors coming from the matrix ( a AB ). Functions like σ ⊥ i ( · ) from definition 2appear every time B i = 2 n + 1, for some i , and this happens in all terms except in the coefficientof ω ∧ · · · ∧ ω n , which is accompanied by the elementary symmetric functions of ( a ij ). Finally,it is a matter of separating those minors according to the 2 n -form which multiplies them. (cid:3) Development towards demonstrating theorems A and B Poincar´e index. Let S nθ be a parallel of latitude θ ∈ ( − π , π ) and let ι = ι θ : S nθ → M be its natural embedding. We may assume that p belongs to the northern hemisphere of S n +1 ,while − p is in the southern hemisphere. Given (cid:15) > S n π − (cid:15) is a small parallel near p , andtogether with S nθ we have an associated annulus region A nθ, (cid:15) of dimension 2 n , with boundary S n π − (cid:15) ∪ S nθ ; see figure 1. FABIANO G. B. BRITO, ANDR´E O. GOMES, AND ICARO GONC¸ ALVES Figure 1. S n +1 with an annulus region near north pole.By Stokes’ theorem, (cid:90) A nθ, (cid:15) d ι ∗ ( E ( (cid:126)v ⊥ )) = (cid:90) S nπ − (cid:15) ∪ S nθ ι ∗ ( E ( (cid:126)v ⊥ )) . However, E ( (cid:126)v ⊥ ) is closed, so d ι ∗ ( E ( (cid:126)v ⊥ )) = 0 and we conclude that the integrals of its restrictionsto both spheres are equal,(7) (cid:90) S nπ − (cid:15) ι ∗ ( E ( (cid:126)v ⊥ )) = (cid:90) S nθ ι ∗ ( E ( (cid:126)v ⊥ )) . Next we compute the restriction of ι ∗ ( E ( (cid:126)v ⊥ )) on S nθ .We may suppose that e , . . . , e n − are all tangent to S nθ . Let α ∈ [0 , π ] be the orientedangle from the tangent space of S nθ to (cid:126)v . In this case, { e , . . . , e n − , u := sin αe n + cos α(cid:126)v } isan orthonormal positively oriented local frame on S nθ .Fix 0 ≤ k ≤ n . Following equation 6 of Lemma 1, we decompose W ( k ) as follows W ( k ) = n − (cid:88) l =1 σ ⊥ k ( l ) ω ∧ · · · ∧ (cid:99) ω l ∧ · · · ∧ ω n +1 + σ ⊥ k (2 n ) ω ∧ · · · ∧ ω n − ∧ ω n +1 + σ k ω ∧ · · · ∧ ω n . OWER BOUND FOR THE VOLUME OF UNIT VECTOR FIELDS ON PUNCTURED SPHERES 9 By applying W ( k ) on ( e , . . . , e n − , u ), we see that ω ∧ · · · ∧ (cid:99) ω l ∧ · · · ∧ ω n +1 ( e , . . . , e n − , u ) = 0 , when 1 ≤ l ≤ n − 1, because e l is in ( e , . . . , e n − , u ) but ω l is omitted. Thus, just the lasttwo terms remain, i.e., W ( k )( e , . . . , e n − , u ) = sin α σ k + cos α σ ⊥ k (2 n ) . Therefore,(8) ι ∗ ( E ( (cid:126)v ⊥ )) = 2vol( S n ) n (cid:88) k =0 (cid:32) nk (cid:33)(cid:32) n k (cid:33) − Ä sin α σ k + cos α σ ⊥ k (2 n ) ä ν S nθ . Going back to 7, its right hand side remains unchanged when we take the limit as (cid:15) goesto zero. Nevertheless, its left hand side is an integral of a function similar to the one appearingin 8, but for a different angle, since this angle depends on latitude of the parallel S n π − (cid:15) , andof course on the vector (cid:126)v . Thus, as (cid:15) goes to zero the only non-vanishing term comes fromthe restriction of (cid:126)v to S n π − (cid:15) , which is the degree of (cid:126)v : S n π − (cid:15) → S n , and this degree equals thePoncar´e index around p (cf. [7]). Therefore,(9) lim (cid:15) → (cid:90) S nπ − (cid:15) ι ∗ ( E ( (cid:126)v ⊥ )) = I (cid:126)v ( p ) . Following a similar argument,(10) lim (cid:15) → (cid:90) S n − π (cid:15) ι ∗ ( E ( (cid:126)v ⊥ )) = I (cid:126)v ( − p ) . Inequalities: volume of a matrix. Our previous discussion determines how the Eulerform relates to the volume form of S nθ , and when the Poincar´e indices of (cid:126)v arise when arepresentative of the Euler class of (cid:126)v ⊥ restricts to small neighborhoods around ± p . Now wecompare the function on 8 to » det(Id + ( ∇ (cid:126)v ) ∗ ( ∇ (cid:126)v )).Following [5], the volume of a linear transformation T : V m → V m is the volume of thegraph of the cube under T . Equivalently, Proposition 1 ([5]) . Let T be an endomorphism and B = ( b ij ) the matrix of T associated tosome orthonormal basis. Then vol( T ) = Ñ (cid:88) ≤ i,j ≤ m b ij + (cid:88) i
Lemma 2. According to the notation settled above, (12) » det(Id + ( ∇ (cid:126)v ) ∗ ( ∇ (cid:126)v )) ≥ n (cid:88) k =0 (cid:32) nk (cid:33)(cid:32) n k (cid:33) − Ä | σ k | + | σ ⊥ k (2 n ) | ä . OWER BOUND FOR THE VOLUME OF UNIT VECTOR FIELDS ON PUNCTURED SPHERES 11 Proof. We define a (2 n + 2) × (2 n + 2) matrix ( b AB ) by adding to ( a AB ) a column and a row ofzeros, ( b AB ) = ( a ij ) a n +1 a n n +1 · · · · · · , so vol( b AB ) = vol( a AB ) = » det(Id + ( ∇ (cid:126)v ) ∗ ( ∇ (cid:126)v )) . By changing the basis, we can write ( b AB ) as a upper triangular matrix, having its eigen-values in the main diagonal (some of them possibly complex)( b AB ) = λ ∗ · · · · · · ∗ λ r ∗ · · · x − y ∗ · · · y x ∗ ... x s − y s · · · · · · y s x s . In general, ( a ij ) is not a symmetric matrix, since (cid:126)v ⊥ is not necessarily integrable. Thus,even though ( b AB ) is possible a non-diagonal matrix, it has at least two zero eigenvalues, say λ and λ , and this fact plays a role when counting its elementary symmetric functions. If wedefine D = diagonal(0 , , | λ | , . . . , . . . , | λ r | , » x + y , » x + y , . . . , » x s + y s , » x s + y s ), then11 holds for this diagonal matrix. Summation goes up to n instead of n + 1 simply because D is equivalent to a 2 n × n matrix. The fact that ( b AB ) has elements above its main diagonalimplies that vol( b AB ) ≥ vol( D ). Since D has nonnegative entries, σ k ( D ) ≥ σ k (( b AB )) (cf. [5], Sections 3 and 4). Therefore omitting the symmetric functions σ ⊥ k ( l ), for 1 ≤ l ≤ n − b AB ) ≥ n (cid:88) k =0 (cid:32) nk (cid:33)(cid:32) n k (cid:33) − σ k ( b AB ) ≥ n (cid:88) k =0 (cid:32) nk (cid:33)(cid:32) n k (cid:33) − Ä σ k + σ ⊥ k (2 n ) ä . (cid:3) Proof of theorem A. We split the integral 1 on M as an integral on a parallel S nθ oflatitude θ ∈ ( − π , π ), and a integral on θ itself,vol( (cid:126)v ) = (cid:90) M » det(Id + ( ∇ (cid:126)v ) ∗ ( ∇ (cid:126)v )) ν M = (cid:90) π − π Ç (cid:90) S nθ » det(Id + ( ∇ (cid:126)v ) ∗ ( ∇ (cid:126)v )) ν S nθ å dθ. From equation 12,vol( (cid:126)v ) ≥ n (cid:88) k =0 (cid:32) nk (cid:33)(cid:32) n k (cid:33) − (cid:90) π − π Ç (cid:90) S nθ Ä | σ k | + | σ ⊥ k (2 n ) | ä ν S nθ å dθ Since sin and cos are bounded, n (cid:88) k =0 Ä nk äÄ n k ä Ä sin α σ k + cos α σ ⊥ k (2 n ) ä ≤ n (cid:88) k =0 Ä nk äÄ n k ä | σ k | + n (cid:88) k =0 Ä nk äÄ n k ä (cid:12)(cid:12)(cid:12) σ ⊥ k (2 n ) (cid:12)(cid:12)(cid:12) , and then, from equations 8 and 7,vol( (cid:126)v ) ≥ vol( S n )2 (cid:90) π − π (cid:90) S nθ ι ∗ ( E ( (cid:126)v ⊥ )) = vol( S n )2 Ñ (cid:90) − π (cid:90) S n − π (cid:15) ι ∗ ( E ( (cid:126)v ⊥ )) + (cid:90) π (cid:90) S nπ − (cid:15) ι ∗ ( E ( (cid:126)v ⊥ )) é Therefore, vol( (cid:126)v ) ≥ π S n ) ( | I (cid:126)v ( p ) | + | I (cid:126)v ( − p ) | ) , which proves theorem A.3.4. A modest extension to arbitrary isolated singularities: proof of theorem B. Forevery p i , 1 ≤ i ≤ m , we can take the exponential map on T p i M and find a real number θ i such that a geodesic sphere S nθ i is the boundary of a geodesic ball in M n +1 , centered in p i andcontaining one singularity, namely p i .Given (cid:15) i > θ i , we build an annulus region A nθ i , (cid:15) i of dimension 2 n , withboundary S n(cid:15) i ∪ S nθ i . Figure 2 illustrates the idea when we restrict ourselves to the case M = S n +1 . We proceed as in subsection 3.1. OWER BOUND FOR THE VOLUME OF UNIT VECTOR FIELDS ON PUNCTURED SPHERES 13 Figure 2. A sphere with various isolated points, each one having a small annulusregion around it.We merely consider that (cid:90) M » det(Id + ( ∇ (cid:126)v ) ∗ ( ∇ (cid:126)v )) ≥ (cid:88) i (cid:90) S nθi » det(Id + ( ∇ (cid:126)v ) ∗ ( ∇ (cid:126)v ))In this case, inequality 12 still holds. Therefore,vol( (cid:126)v ) ≥ vol( S n )2 m (cid:88) i =1 (cid:90) S nθi ι ∗ ( E ( (cid:126)v ⊥ )) = vol( S n )2 m (cid:88) i =1 | I (cid:126)v ( p i ) | Concluding remarks Even though, compared to theorem A, the lower bound found in 3 is not sharp when m = 2and M = S n +1 \{± p } , it presents a lower value for vector fields having two singularities in arandom position, rather than on antipodal points.Additionally, as discussed in [6] for the energy functional, given a number (greater thantwo) of isolated singularities, it is possible to find a unit vector field having these singularitiesand with volume arbitrarily close to the volume of the radial vector field. This may be doneby the following argument: put two singularities in antipodal points ± p and every remainsingularity in a neighborhood near the south pole − p , for example. Outside this neighborhood,take the radial vector field coming from p and inside it one can take any vector field preserving the indices that were established in the beginning. By gluing those two parts together, one canobtain a vector field such that its volume is close to the volume of V R . This is possible sincethe smaller the neighborhood, the smaller the volume.Theorem B represents a fair topological step towards a more general geometric question:is it possible to determined a unit vector field of minimum volume on a Riemannian manifoldwithout a subset of singularities in a fixed configuration? References [1] Borrelli V., Gil-Medrano O.: Area-minimizing vector fields on round 2-spheres. J. reine angew. Math. ,85–99 (2010)[2] ———: A critical radius for unit Hopf vector fields on spheres. Math. Ann. (4), 731–751 (2006)[3] Brito, F.G.B., Chac´on, P.M.: A topological minorization for the volume of vector fields on 5-manifolds.Arch. Math. , 283–292 (2005)[4] Brito, F.G.B., Chac´on, P.M., Johnson, D.L.: Unit vector fields on antipodally punctured spheres: Bigindex, big volume. Bull. Soc. Math. Fr. (1), 147–157 (2008)[5] Brito, F.B., Chac´on, P.M., Naveira, A.M.: On the volume of unit vector fields on spaces of constant sectionalcurvature. Comment Math. Helv. , 300–316 (2004)[6] Chac´on, P.M., Nunes G. S.: Energy and topology of singular unit vector fields on S . Pacific J. Math. (1), 27–34 (2007)[7] Chern, S.S.: A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds. Ann.of Math. (4), 747–752 (1944)[8] Gluck, H., Ziller, W.: On the volume of a unit field on the three-sphere. Comment Math. Helv. , 177–192(1986)[9] Johnson, D.L.: Volume of flows. Proc. Amer. Math. Soc. , 923–932 (1988)[10] Pedersen, S.L.: Volume of vector fields on spheres. Trans. Amer. Math. Soc. , 69–78 (1993)[11] Reznikov, A.G.: Lower bounds on volumes of vector fields, Arch. Math. , 509–513 (1992) Centro de Matem´atica, Computac¸˜ao e Cognic¸˜ao, Universidade Federal do ABC, 09.210-170Santo Andr´e, Brazil E-mail address : [email protected] OWER BOUND FOR THE VOLUME OF UNIT VECTOR FIELDS ON PUNCTURED SPHERES 15 Dpto. de Matem´atica, Instituto de Matem´atica e Estat´ıstica, Universidade de S¯ao Paulo,R. do Mat¯ao 1010, S¯ao Paulo-SP 05508-900, Brazil. E-mail address : [email protected] Dpto. de Matem´atica, Instituto de Matem´atica e Estat´ıstica, Universidade de S¯ao Paulo,R. do Mat¯ao 1010, S¯ao Paulo-SP 05508-900, Brazil. E-mail address ::