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2. Proof of the main result One of the most important techniques in Riemannian geometry is the Bochner technique [16, Chapter 9], [22]. Roughly, it consists of relating the existence of certain non-trivial tangent vector fields on a Riemannian manifold with properties of the curvature of such a manifold. In the compact case, it leads to the vanishing of certain geometrically relevant tangent vector fields (Killing or conformal) under the assumption of positive or negative curvature everywhere [22]. Our argument for the proof of a b in Theorem 1 will split into three steps. In the first one, given a 2-dimensionalp qñp Riemannianq manifold M,g , if it admits a smooth tangent vector field with no zeroes, we use the well knownp Bochnerq formula (see, e.g., [22, 19] ) to construct a tangent vector field on M whose divergence, with respect to g, is equal to the Gauss curvature of g. In the second step, we will start from a compact and connected 2-dimensional smooth manifold M which admits a nowhere zero continuous tangent vector field. We will endow M with a Riemannian metric g. Then we will prove, by a classical approximation argument, that existence of the nowhere zero continuous tangent vector field leads to the existence of a smooth one also with no zeroes. Finally, in the last step, we will make use of the global Gauss-Bonnet theorem [1, 8] to get that M must be a torus. Step 1. An application of the Bochner formula Consider a compact, orientable and connected 2-dimensional manifold M. The classical Whitney embedding theorem [2, Theorem 5.4.8] guarantees that we can get a smooth Topology of orientable connected and compact surfaces 3 embedding of M into Rn, for n large enough. Then we can consider on M the metric g induced by the usual one of Rn and this is precisely what we do. The Bochner formula states that 2 X div X Ric X,X div ∇X X trace AX , p p qq“´ p q` p q´ p q for all smooth tangent vector field X, where Ric is the Ricci tensor of M,g , div denotes p q the divergence on M,g and AX is the linear operator field defined by AX v ∇vX, p q p q“´ being ∇ the Levi-Civita connection of g and v TxM, x M [22, 19]. P P Now observe that div X trace AX , p q“´ p q for any X and any dimension of M. On the other hand, when dim M 2, we have “ Ric X,Y Kg X,Y , p q“ p q for all X,Y , where K denotes the Gauss curvature of g. 1 We want to apply Bochner formula to the unitary tangent vector field T : T g T, T 2 “ { p q in the case dim M 2. Taking into account g T , T 1, we have that the matrix of AT respect to an orthonormal“ basis T , E is p q“ p q 0 0 . a b ˆ ˙ Therefore, we have 2 2 2 trace AT trace AT div T . p q“p p qq “p p qq Hence, we arrive to the following formula, 2 (1) T div T K div ∇T T div T . p p qq“´ ` p q´p p qq On the other hand, using the well known formula div fX X f fdiv X , p q“ p q` p q which holds true for any tangent vector field X and any smooth function f on M, we get (2) div T 2 div div T T T div T , p p qq “ p q ´ p p qq and substituting (2) in (1), we get ` ˘ K div ∇T T div div T T “ p q´ p p q q

div ∇T T div T T . “ ´ p q ´ ¯

Thus, taking Y ∇T T div T T we have the announced formula “ ´ p q

(3) K div Y . “ p q 4 J. M. Almira, A. Romero

Step 2. The approximation argument Every continuous tangent vector field X on M may be identified with a continuous map n X X1, ,Xn : M R , “p ¨ ¨ ¨ q ÝÑ n satisfying that X x lies in the tangent space Tx M (contemplated into R via the em- bedding), for all xp q M. Now, assume a continuousp q tangent vector field X that vanishes nowhere and considerP X Z : Z1, ,Zn , “ X “p ¨ ¨ ¨ q } } n where denotes the Euclidean norm of R . Note that every function Zi : M R, 1 i }¨}n, is continuous. Ñ ď ď On the other hand, compactness of M implies that n M B0 r x R : x r , Ă p q “ t P } }ď u for some r 0. Thus, Tietze’s extension theorem [11, 14] implies that there exist functions ą Fi C B0 r , R such that Fi M Zi for i 1, , n. P p p q q | “ “ ¨ ¨ ¨ Apply now the Stone-Weierstrass approximation theorem [17, Theorem 4.1] (see also [20]) to the space of continuous functions C B0 r , R and its subalgebra R x1, , xn p p q q r ¨ ¨ ¨ s to conclude that there are polynomials Pi R x1, , xn , 1 1 n, such that P r ¨ ¨ ¨ s ď ď 1 sup Fi x Pi x , i 1, , n, }x}ďr | p q´ p q| ă 2?n “ ¨ ¨ ¨ which implies that

1 1 n 2 n 2 2 2 sup Zi x Pi x sup Fi x Pi x x M x r P ˜i“1 | p q´ p q| ¸ ď } }ď ˜i“1 | p q´ p q| ¸ ÿ ÿ 1 2 2 n max sup Fi x Pi x 1 i n ď ˜ p ď ď }x}ďr | p q´ p q|q ¸ 1 1 2 1 1 n . ă 4n “ 4 “ 2 ˆ ˙ c In particular, n P P1, ,Pn : M R , “p ¨ ¨ ¨ q ÝÑ defines a smooth map. Let us now consider the tangential component T of P . In other n n words, we consider, for each x M, the decomposition of TxR as a direct sum TxR K P “ TxM Tx M and decompose ‘ P T U, “ ` Topology of orientable connected and compact surfaces 5

K along the embedding, with T x TxM and U x Tx M, for each x M. It follows p q P p q P P that T x 0 for all x M since T x0 0 for some x0 M implies that p q‰ P p q“ P 1 2 2 2 P x0 Z x0 U x0 Z x0 U x0 1 1, 4 ą} p q´ p q} “} p q´ p q} “} p q} ` ą which is impossible (note that in the formula above we have used that U x0 Z x0 ). Hence T is a smooth tangent vector field with no zeroes on M. p q K p q Remark. Obviously, the same conclusion can be obtained for M of any dimension. Step 3. The Gauss-Bonnet argument If we apply the classical divergence theorem taking in mind (3), we have that

K dA 0, M “ ż (compare with [8, Theorem p. 68]). However, from the Gauss-Bonnet theorem we know

K dA 2πχ M , M “ p q ż where χ M is the Euler number of M. Therefore, χ M 0 and consequently M is a torus [9,p Theoremq 9.3.11]. p q “

3. Acknowledgement We want to express our acknowledgement to the referee, since his/her many suggestions have deeply improved the clarity of our arguments, making the paper more transparent and self-contained.

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[10] T. Jarvis, J. Tanton, The Hairy Ball Theorem via Sperner’s Lemma, American Mathematical Monthly 111 (7) (2004) 599–603. [11] M. Mandelkern, A short proof of the Tietze-Urysohn extension theorem, Arch. Math. 60 (1993) 364–366. [12] J. W. Milnor, Topology from the Differentiable Viewpoint, University of Virginia Press, Char- lottesville, 1965. [13] J. W. Milnor, Analytic proofs of the “hairy ball theorem” and the Brouwer fixed point theorem, Amer. Math. Monthly 85 (1978) 521–524. [14] E. Ossa, A simple proof of the Tietze-Urysohn extension theorem, Arch. Math. 71 (1998) 331–332. [15] J. Parisi, J. Ball, The Future of Fusion Energy, World Scientific, 2019. [16] P. Petersen, Riemannian Geometry, Graduate texts in Math. 171, Springer Verlag 2006. [17] A. Pinkus, Density in approximation theory, Surveys in Approximation Theory 1 (2005) 145. [18] H. Poincare, Sur les courbes definies par les equations differentielles, J. Math. Pures Appl. (4) 1 (1885) 167–244. [19] A. Romero, The introduction of Bochner’s Technique on Lorentzian Manifolds, Nonlinear Anal. 47 (2001) 3047–3059. [20] M. H. Stone, A generalized Weierstrass approximation theorem, Math. Magazine 21 (4) (1948) 167-184 and 21 (5) (1948) 237–254. [21] E. F. Whittlesey, Fixed points and antipodal points, Amer. Math. Monthly 70 (8) (1963) 807– 821. [22] H. Wu, The Bochner technique in Differential Geometry, Classical Topics in Mathematics, 6. Higher Education Press, Beijing, 2017.

AMS Subject Classification: 53C20, 53C23, 57R25

Keywords: Bochner technique, Orientable compact connected surfaces, Gauss curvature, Gauss-Bonnet theorem

J. M. ALMIRA, Departamento de Ingenier´ıay Tecnolog´ıade Computadores, Universidad de Murcia. 30100 Murcia, SPAIN e-mail: [email protected]

A. ROMERO, Departamento de Geometr´ıay Topolog´ıa, Universidad de Granada 18071 Granada, SPAIN. e-mail: [email protected]