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A Geodesic Connection in Fréchet Geometry

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A Geodesic Connection in Fréchet Geometry A geodesic connection in Fr´echet Geometry Ali Suri Abstract. In this paper first we propose a formula to lift a connection on M to its higher order tangent bundles T rM, r 2 N. More precisely, for a given connection r on T rM, r 2 N [ f0g, we construct the connection rc on T r+1M. Setting rci = rci−1 c, we show that rc1 = lim rci exists − and it is a connection on the Fr´echet manifold T 1M = lim T iM and the − geodesics with respect to rc1 exist. In the next step, we will consider a Riemannian manifold (M; g) with its Levi-Civita connection r. Under suitable conditions this procedure i gives a sequence of Riemannian manifolds f(T M, gi)gi2N equipped with ci c1 a sequence of Riemannian connections fr gi2N. Then we show that r produces the curves which are the (local) length minimizer of T 1M. M.S.C. 2010: 58A05, 58B20. Key words: Complete lift; spray; geodesic; Fr´echet manifolds; Banach manifold; connection. 1 Introduction In the first section we remind the bijective correspondence between linear connections and homogeneous sprays. Then using the results of [6] for complete lift of sprays, we propose a formula to lift a connection on M to its higher order tangent bundles T rM, r 2 N. More precisely, for a given connection r on T rM, r 2 N [ f0g, we construct its associated spray S and then we lift it to a homogeneous spray Sc on T r+1M [6]. Then, using the bijective correspondence between connections and sprays, we derive the connection rc on T r+1M from Sc. Setting rci = rci−1 c, we show that rc1 = lim rci exists and it is a connection on the Fr´echet manifold T 1M = lim T iM − − and the geodesics with respect to rc1 exist. In the next step, we will consider a Riemannian manifold (M; g) with its Levi- Civita connection r. Using the results of the previous section, we construct the c1 c 1 connection r := r and the Sasaki Metric g1 on T M = TM. It is known that if rc0 := r is a flat connection then, rc1 is a flat connection too and rc1 is the 1 Levi-Civita connection of the Riemannian metric g1 on T M. Balkan∗ Journal of Geometry and Its Applications, Vol.23, No.2, 2018, pp. 67-75. c Balkan Society of Geometers, Geometry Balkan Press 2018. 68 Ali Suri i Repeating this procedure gives a sequence of Riemannian manifolds f(T M, gi)gi2N ci equipped with a sequence of Riemannian connections fr gi2N. Note that the limit connection rc1 = lim rci is a generalized linear connection on the non-Banach man- − ifold T 1M. The advantage of using this procedure lies in the fact that rc1 produces the curves which are the (local) length of T 1M and hence we call it a geodesic connection. In this paper we assume that all the maps and manifolds are smooth and when the partition of unity is necessary we suppose that our manifolds admit partition of unity. 2 Preliminaries Let M be a smooth manifold modeled on the Banach space E and π0 : TM −! M S be its tangent bundle. We remind that TM = x2M TxM such that TxM consists of all equivalence classes of the form [c; x] where c 2 Cx = fc :(, ) −! M; > 0; c is smooth and c(0) = xg; under the equivalence relation 0 0 c1 ∼x c2 () c1(0) = c2(0) for c1; c2 2 Cx. The projection map π0 : TM −! M maps [f; x] onto x. If A0 = f(φα0 := φα;Uα0 := Uα); α 2 Ig is an atlas for M, then we have the canonical atlas −1 A1 = f φα1 := Dφα0 ;Uα1 := π0 (Uα) ; α 2 Ig for TM where −1 φα1 : π0 (Uα0 ) −! −! Uα0 × E 0 [c; x] 7−! ((φα0 ◦ c)(0); (φα0 ◦ c) (0)): Inductively one can define an atlas for T rM := T (T r−1M), r 2 N, by −1 Ar := f φαr := Dφar−1 ;Uαr := πr−1(Uαr−1 ) ; α 2 Ig r r−1 for which πr−1 : T M −! T M is the natural projection. The model spaces r 2 2r for TM and T M are E1 := E and Er := E respectively. We add here to the convention that T 0M = M. r Set κ1 := idTM and for r ≥ 2 consider the canonical involution κr : T M −! r 2 T M which satisfies @t@sf(t; s) = κr@s@tf(t; s), for any smooth map f :(−, ) −! r−2 r−2 T M. If we consider T M as a smooth manifold modeled on Er−2, then the r−1 r r+1 2 4 charts of T M, T M and T M take their values in Er−1 = Er−2, Er = Er−2 8 and Er+1 = Er−2 respectively. It is easy to check that the local representation of κr is given by κ := φ ◦ κ ◦ φ−1 : U × 3 −! U × 3 rα αr r αr αr−2 Er−2 αr−2 Er−2 (x; y; X; Y ) 7−! (x; X; y; Y ): For r ≥ 1 a semispray on T r−1M is a vector field S : T rM −! T r+1M with the additional property κr+1 ◦ S = S (or equivalently Dπr−1 ◦ S = idT r M ) ([1]). A geodesic connection in Fr´echet Geometry 69 Considering the atlas Ar−1, we observe that locally on the chart φar−1 ;Uαr−1 the −1 local representation of the vector field S is Sα := φαr+1 ◦ S ◦ φαr and 3 Sα : Uαr−1 × Er−1 −! Uαr−1 × Er−1 ;(x; y) 7−! x; y; y; −2Gα(x; y) : where Gα : Uαr−1 × Er−1 −! Er−1 locally represents S (see e.g. [1, 6]). Definition 2.1. A semispray S on T rM is called a (2-homogeneous) spray if for any 2 α 2 I, λ 2 R and (x; y) 2 Uαr × Er, Gα(x; λy) = λ Gα(x; y). Definition 2.2. The complete lift of the semispray (spray) S is defined by c (2.1) S = Dκr+2 ◦ κr+3 ◦ DS ◦ κr+2: According to propositions 3.5 and 4.13 of [6], the complete lift of a semispray (spray) is a semispray (respectively spray). 3 Complete lift of connections In this section we compute the iterated complete lifts of a linear connection r on M to a linear connection rci on T iM for i 2 N [ f1g. To this end, first we establish a bijective correspondence between sprays and linear connections. Then, using the results of [6] we will lift a connection to its higher order tangent bundles. The first result is known due to Lang [4]. However, in order to make our exposition as much as possible self contained, we state a proof adopted to our terminology. First, we recall the definition of a connection from [7]. Definition 3.1. A connection on M is a vector bundle morphism r : TTM −! TM with the family of local components (Christoffel symbols) fΓαgα2I where; Γα : φ(Uα) × E !L(E; E); α 2 I; −1 and the local expression of r, i.e. rα = φα1 ◦ r ◦ φα2 is 3 rα : φα(Uα) × E −! φα(Uα) × E (x; y; X; Y ) 7−! (x; Y + Γα(x; y)X): If the local components fΓαgα2I are linear with respect to their second variable (and 2 symmetric i.e. Γα(x)(y; X) = Γα(x)(X; y) for all (y; X) 2 E ), then the connection is called a linear (and symmetric) connection. Let γ :(−, ) −! TM be a curve. Then γ is a called a geodesic of r if 0 00 0 rγ0(t)γ (t) := r ◦ γ (t) = 0. The local representation of rγ0(t)γ (t) = 0 is 00 0 0 (3.1) γα(t) + Γα(γα(t)) γα(t); γα(t) = 0; t 2 (−, ); α 2 I: Proposition 3.1. i. Any spray S on M gives rise to a linear and symmetric con- nection r on M and vice versa. ii. The curve γ :(−, ) −! M is a geodesic of the spray S if and only if γ is a geodesic of r. 70 Ali Suri Proof. Suppose that S be a spray on M with the local expression −1 Sα := φα2 ◦ S ◦ φα1 : TM −! TTM (x; y) 7−! x; y; y; −2Gα(x; y) where Gα; α 2 I; is a 2-homogeneous function with respect to its second variable. 1 Set B (x) = @2G (x; 0). Clearly B(x) 2 L2 ( ; ). According to [4] (chapter I, α 2 2 α sym E E 1 2 section 3) Gα(x; y) = 2 @2 Gα(x; 0)(y; y) for any (x; y) 2 Uα × E. Define the local components of the connection r on M by Γα(x)(y; y) = 2Bα(x)(y; y) = 2Gα(x; y). Then for (y; z) 2 E × E we have 1 Γ (x)(y; z) = fΓ (x)(y + z; y + z) − Γ (x)(y; y) − Γ (x)(z; z)g: α 2 α α α 2 Clearly Γα : Uα ⊂ M −! Lsym(E; E), α 2 I, is smooth. Then the connection map, again denoted by r, is define by rα : TTMjUα −! TMjUα (x; y; X; Y ) 7−! x; Y + Γα(x)(y; X) : We will show that Γα and Γβ satisfy the known compatibility condition 2 dφαβfΓβ(x)(y; z)g = d φαβ(x)(y; z)(3.2) +Γα φαβ(x) [dφαβ(x)y; dφαβ(x)z]: (see e.g.
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