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A Note on Locally Metric Connections Mathematics and Statistics 7(4): 146-149, 2019 http://www.hrpub.org DOI: 10.13189/ms.2019.070408 A Note on Locally Metric Connections Mihail Cocos Department of Mathematics, Weber State University, Ogden, UT 84408, USA Received June 17, 2019; Revised August 30, 2019; Accepted September 15, 2019 Copyright c 2019 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract The Fundamental Theorem of Riemannian ifold is related to a long outstanding conjecture of Chern geometry states that on a Riemannian manifold there ex- for affinely flat manifolds. Chern's statement conjectures ist a unique symmetric connection compatible with the that the Euler characteristic of a compact afinely flat man- metric tensor. There are numerous examples of connec- ifold is zero. The author showed in ([5]) that for a locally tions that even locally do not admit any compatible met- metric connection there is a natural cohomology class of rics. A very important class of symmetric connections M that coincides with the Euler class of the bundle E in in the tangent bundle of a certain manifolds (afinnely the case when the connection is globally metric. He also flat) are the ones for which the curvature tensor vanishes. proved that if the set of locally metric connections in the Those connections are locally metric. S.S. Chern con- tangent bundle of an affinely flat manifold is path con- jectured that the Euler characteristic of an affinely flat nected, then Chern's statement is true(see [5]). manifold is zero. A possible proof of this long outstand- If two symmetric and locally metric connections in the ing conjecture of S.S. Chern would be by verifying that tangent bundle of a manifold M share a parallel metric in the space of locally metric connections is path connected. the neighborhood of any point then, by the fundamental In order to do so one needs to have practical criteria for theorem of Riemannain geometry, they are equal. If the the metrizability of a connection. In this paper we give symmetry is dropped one can no longer conclude that they necessary and sufficient conditions for a connection in a are equal and the relationship becomes a weaker equiva- plane bundle above a surface to be locally metric. These lence relation. The global result of this paper proves that conditions are easy to be verified using any local frame. two locally metric equivalent connections have the same Also, as a global result we give a necessary condition for Euler class. two connections to be metric equivalent in terms of their Euler class. 2 On the local metrizability of connections Keywords Affine Connections, Locally Metric Connec- tions, Affinely Flat Manifolds, Euler Class of a Connection We will now establish some conditions for a connection to be metric in terms of local frames. Lemma 2.1. Let D be connection in E: Then D is locally metric if and only if in the neighborhood of any point there 1 Introduction exist a local frame of the bundle Throughout this paper E will denote a real vector bun- σ = (σ1; σ2; ··· ; σm) dle of rank m over a manifold M of dimension n: such that the connection matrix θ with respect to σ is skew Definition 1.1. A connection D in E is called locally symmetric. metric if and only if there is a bundle metric that is par- allel with respect to D in the neighborhood of any point Proof. The only if part is obvious. For the if part let p 2 M: σ = (σ1; σ2; ··· ; σm) Equivalently one can prove that a connection is locally as in hypothesis. Take the metric g that makes the frame metric if its restricted holonomy group Hol0(D) is com- pact( see [3]). However, since the explicit calculation σ orthonormal, that is of holonomy is very difficult, we will not make use of this variant of the definition. The literature for locally g(σi; σj) = δij: metric connections is scarce. There are not very many Differentiating g in the direction X 2 T M; we get practical criteria to verify weather a connection is lo- cally metric. Some promising results can be found in (DX g)(σi; σj) = 0 − (θij(X) + θji(X)) = 0; ([1, 2, 9, 10, 8, 4, 6]) From a pure mathematical point of view the investigation of the set of symmetric and lo- hence g is parallel with respect to D: cally metric connections in the tangent bundle of a man- Mathematics and Statistics 7(4): 146-149, 2019 147 For the sake of concreteness let us give a couple of ex- follows immediately from Cartan's equation amples of connections which are locally metric but not globally metric. Let M = T2 be the two dimensional ! = dθ + θ ^ θ torus and let f = (f ; f ) a global frame of commuta- 1 2 by taking the trace of both sides. tive vector fields in T M: The dual frame will be denoted Next let ! be a volume form on E: Whitout loss of gener- f ∗ = (f 1; f 2): Consider the connection D that has its ality we may assume that the frame σ = (σ ; σ ; ··· ; σ ) connection matrix with respect to f 1 2 m is positive with respect to !; that is: 1 f 0 1 2 m θf = ! = fσ ^ σ ^ ::::: ^ σ ; (2) 0 0 This is a flat, symmetric connection in TM hence locally where f > 0 is a local smooth function and σ∗ = metric. However there is no global metric on M that will (σ1; σ2; ··· ; σm) is the dual frame of σ:. Taking the deriva- have D as its Levi Civita connection. Assume, by contra- tive of ! in the direction of an arbitrary tangent vector diction, that there is a global metric g that is preserved by field X we get D and consider γ = γ(t); −∞ < t < 1 an integral curve m of f : On M we have a globally defined smooth function 1 2 m X 1 2 k m 1 DX ! = X(f)σ ^σ :::^σ +f σ ^σ ^::::^DX σ :::^σ defined as k=1 h = g(f1; f1); (3) Since and if we look at its restriction h(t) to γ, then h(t) satisfies k s the differential equation DX e = −θsk(X)e and by using (3) we obtain h0(t) = 2h(t); m X 1 2 m and hence it is an exponential. Consequently h is not DX ! = (X(f) − f θkk(X))σ ^ σ ::: ^ σ (4) bounded on M which is a contradiction. k=1 which shows that Before giving the second example we need to make a D! = 0 definition is equivalent to Definition 2.1. Let D1 and D2 be two locally metric con- df = fT r θ nections in a vector bundle E: We say they are metric equivalent if and only if in the neighborhood of any point or there exist a local bundle metric g such that d(ln f) = T r θ: (5) Obviously (5) is equivalent to T r θ is closed. D1g = D2g = 0: For the case of plane bundles we will be able to give a ~ Our second example is a connection r in the tangent bun- practical test for local metrizability. For this criteria we dle of a generic Riemannian manifold (M; g) that is metric will need the following linear algebra lemmas equivalent (rg ≡ r~ g ≡ 0) to its Levi Civita connection r but not symmetric. We will base our example on The- Lemma 2.3. Let U be a nonsingular 2 × 2: Let A; B two orem 2.1 in ([7]). Take u a one form on M and let u# be matrices with determinant equal to one that satisfy its metric dual with respect to g: Define A−1UA = B−1UB = kJ; ~ # rX Y = rX Y + u(Y )X − G(X; Y )u (1) 0 1 with J = : Then It Is easy to verify that equation (1) defines a connection −1 0 compatible with the metric g and that its torsion satisfies B = AS; T ~ (X; Y ) = u(Y )X − u(X)Y; r with S orthonormal. and therefore r~ is non-symmetric if u 6= 0: Proof. First, let us note the following easy to prove iden- tity for 2 × 2 matrices Next let us note that if a connection preserves a metric (locally) then it also preserves a volume form, hence we XJX−1 = XXT J; (6) have the following criterion where X is any 2 × 2 matrix with determinant equal to Lemma 2.2. Le D be a connection in a bundle E and one and J is as in the hypothesis. Using the hypothesis θ be its connection matrix with respect to the local frame we get σ = (σ1; σ2; ··· ; σm): Then D preserves a local volume U = kAJA−1 = kBJB−1; form if and only if and by using (6) we get T r Ω = d(T r θ) = 0: AAT = BBT (7) Proof. First let us note that and therefore T r Ω = d(T r θ) A−1B = AT (BT )−1: (8) 148 A Note on Locally Metric Connections Let's set Proof. For the "only if" part, since D is assumed to be S = A−1B = AT (BT )−1: locally metric, according to Lemma there exist a frame τ such that the connection matrix with respect to this We have frame is skew and consequently its curvature matrix Ψ is also skew.
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