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Automorphisms of Ehresmann Connections Acta Math. Hungar., 123 (4) (2009), 379395. DOI: 10.1007/s10474-008-8139-x First published online December 12, 2008 AUTOMORPHISMS OF EHRESMANN CONNECTIONS J. PÉK and J. SZILASI¤ Institute of Mathematics, University of Debrecen, H-4010 Debrecen, P.O.B. 12, Hungary e-mail: [email protected] (Received July 14, 2008; accepted July 28, 2008) Abstract. We prove that a dieomorphism of a manifold with an Ehresmann connection is an automorphism of the Ehresmann connection, if and only if, it is a totally geodesic map (i.e., sends the geodesics, considered as parametrized curves, to geodesics) and preserves the strong torsion of the Ehresmann connection. This result generalizes and to some extent strengthens the classical theorem on the automorphisms of a D-manifold (manifold with covariant derivative). 1. Introduction It is well-known (and almost trivial) that two covariant derivative oper- ators on a manifold are equal, if and only if, they have the same geodesics and equal torsion tensors. Heuristically, this implies that a dieomorphism of a manifold with a covariant derivative is an automorphism of the covari- ant derivative, if and only if, it is a totally geodesic map (i.e. sends geodesics, considered as parametrized curves, to geodesics) and preserves the torsion. ¤The second author is supported by the Hungarian Nat. Sci. Found. (OTKA), Grant No. NK68040. Key words and phrases: Ehresmann connection, automorphism, strong torsion, totally geodesic map. 2000 Mathematics Subject Classication: 53C05, 53C22. 02365294/$ 20.00 °c 2008 Akadémiai Kiadó, Budapest 380 J. PÉK and J. SZILASI A formal proof of this conceptually important result may be found e.g. in a celebrated paper of J. Vilms [14] for the torsion-free case. Here we shall investigate similar questions for manifolds endowed with an Ehresmann connection. Briey, by an Ehresmann connection over a man- ifold M we mean a bre preserving, brewise linear map from TM £M TM into TTM, which is a right inverse of the canonical surjection TTM ! ± ± TM £M TM and smooth over TM £M TM, where TM is the bundle of the nonzero tangent vectors to M (confer C1C4 in Section 6). In his in- uential paper [7], J. Grifone introduced the concept of strong torsion of an Ehresmann connection and proved that an Ehresmann connection is uniquely determined by its strong torsion and geodesics. This result makes it plausi- ble that a totally geodesic map of a manifold with an Ehresmann connection which preserves the strong torsion is an automorphism of the Ehresmann connection (and conversely). After quite detailed, but not too dicult prepa- rations we show that this is indeed true. Surprisingly or not, a large part of our reasoning is based on simple commutation relations concerning the push-forward operations by a dieomorphism and the fundamental canoni- cal objects introduced in Section 3 on the one hand (Section 5), and some non-canonical commutation relations implied by an automorphism of an Ehresmann connection (Lemmas 7.17.3) on the other hand. In the concluding section we show that if the Ehresmann connection is homogeneous and of class C1 on its whole domain, and hence it canonically leads to a covariant derivative on the base manifold, then our theorem reduces to the classical theorem on totally geodesic maps. More precisely, it turns out that the torsion condition of the classical theorem may be weakened to some extent a phenomenon, which is hidden for the traditional approach. 2. Preliminaries We follow the notation and conventions of [11] (see also [10] and [12]) as far as feasible. However, for the readers' convenience, in this section we x some terminology and recall some basic facts. Manifold will always mean a connected smooth manifold of dimension n 2 N¤ which is Hausdor and has a countable basis of open sets. If M is a manifold, C1(M) will denote the ring of smooth functions on M and Di® (M) the group of dieomorphisms from M onto itself. ¿ : TM ! M (simply, ¿ or TM) is the tangent bundle of M. ¿TM denotes the canonical projection, the foot map, of TTM onto TM, as well as the tangent bun- dle of TM. If ' : M ! N is a smooth map, then '¤ will denote the smooth map of TM into TN induced by ', the tangent map or derivative of '. The vertical lift of f 2 C1(M) is f v := f ± ¿, the complete lift f c 2 C1(TM) of f is dened by f c(v) := v(f), v 2 TM. It follows at once that Acta Mathematica Hungarica 123, 2009 AUTOMORPHISMS OF EHRESMANN CONNECTIONS 381 if ' : M ! M is a smooth map and f 2 C1(M), then c c (1) (f ± ') = f ± '¤: Throughout the paper, I ½ R will be an open interval. The velocity eld of a smooth curve γ : I ! M is d (2) γ_ := γ ± : I ! T M; ¤ du where d is the canonical vector eld on the real line. The acceleration eld du of γ is µ ¶ µ ¶ (2) d d d d (3) γÄ := γ_ = γ¤ ± ± = 㤤 ± ± : du ¤ du du ¤ du If γ : I ! M is a smooth curve and ' 2 Di® (M), then ' ± γ is also a smooth curve, and we have ¡¡¡_ ¡¡¡Ä (4) p' ± γq= '¤ ± γ;_ p' ± γq= '¤¤ ± γ:Ä X(M) denotes the C1(M)-module of smooth vector elds on M. Any vector eld X on M determines two vector elds on TM, the vertical lift Xv of X and the complete lift Xc of X, characterized by (5) Xvf c = (Xf)v;Xcf c = (Xf)c; f 2 C1(M): 3. Canonical constructions © ª ¤ Let ¿ TM := TM £M TM := (u; v) 2 TM £ TM j ¿(u) = ¿(v) , and let ¼(u; v) := u for (u; v) 2 ¿ ¤TM. Then ¼ is a vector bundle with total space ¿ ¤TM and base space TM, the pull-back of ¿ : TM ! M over ¿. The C1(TM)-module of sections of ¼ will be denoted by Sec (¼). Any vector eld X on M determines a section ¡ ¢ Xb : v 2 TM 7¡! v; X ± ¿(v) 2 TM £M T M; called the basic section associated to X, or the lift of X into Sec (¼). The module Sec (¼) is generated by the basic sections. We have a canonical sec- tion ± : v 2 TM 7¡! (v; v) 2 TM £M T M: Acta Mathematica Hungarica 123, 2009 382 J. PÉK and J. SZILASI Generic sections of Sec (¼) will be denoted by X;e Y;:::e . Starting from the slit ± ± ± ± tangent bundle ¿ : TM ! M, the pull-back bundle ¼ : TM £M TM ! TM is constructed in the same way. Omitting the routine details, we remark that ± Sec (¼) may naturally be embedded into the C1(TM)-module Sec (¼± ). There exists a canonical injective bundle map i : TM £M TM ! TTM given by i(u; v) :=c _(0); if c(t) := u + tv (t 2 R); and a canonical surjective bundle map j : TTM ! TM £M T M; w 2 TvTM ¡ ¢ 7¡! j(w) := v; ¿¤(w) 2 fvg £ T¿(v)M: Then j ± i = 0, while J := i ± j is a further important canonical object, the vertical endomorphism of TTM. i and j induce the tensorial maps Xe 2 Sec (¼) 7¡! iXe := i ± Xe 2 X(TM); and » 2 X(TM) 7¡! j» := j ± » 2 Sec (¼); so J may also be interpreted as a C1(TM)-linear endomorphism of X(TM). Xv(TM) := i Sec (¼) is the module of vertical vector elds on TM. The ver- tical vector elds form a subalgebra of the Lie algebra X(TM) at the same time. For any vector eld X on M we have (6) iXb = Xv; jXc = X;b and hence (7) JXv = 0; JXc = Xv: It follows that J2 = 0, Ker (J) = Im (J) = Xv(TM). C := i± is a canonical vertical vector eld, called the Liouville vector eld. If X 2 X(M) then (8) [Xv;C] = Xv: Acta Mathematica Hungarica 123, 2009 AUTOMORPHISMS OF EHRESMANN CONNECTIONS 383 4. Semisprays By a semispray over M we mean a map S : TM ! TTM satisfying the following conditions: SPR1. ¿TM ± S = 1TM . SPR2. jS = ± (or, equivalently, JS = C). ± SPR3. S is smooth on TM. SPR4. S sends the zero vectors of TM to the zero vectors of TTM. A map S : TM ! TTM is said to be a spray if it has the properties SPR1SPR3 and satises the next two conditions: SPR5. [C; S] = S, i.e., S is positive-homogeneous of degree 2. SPR6. S is of class C1 on TM. If S is a spray then the homogeneity condition implies that SPR4 is true automatically. Notice that this concept of a spray was introduced by P. Da- zord in his Thèse [4]. J. Grifone dened a semispray only by conditions SPR1SPR3, see [7]. A spray is said to be an ane spray if it is of class C2 (and hence smooth) on TM. Ane sprays were introduced by Ambrose, Palais and Singer [1] under the name spray. If S is a semispray over M, then for any vector eld » on TM we have (9) J[J»; S] = J»: This useful relation will be cited as Grifone's identity; see [7], and for a simple direct proof [13]. In particular, if X is a vector eld on M, then (10) j[Xv;S] = Xb = jXc; therefore [Xv;S] and Xc dier only in a vertical vector eld. Thus it follows that (11) [Xv;S] = Xc + ´; ´ 2 Xv(TM): By a geodesic of a semispray S we mean a regular smooth curve γ : I ! M that satises the relation (12) γÄ = S ± γ:_ (Regularity excludes the constant geodesics.) A dieomorphism ' 2 Di® (M) is said to be an anity (ane transformation, ane collineation) of a semis- pray S if it preserves the geodesics as parametrized curves.
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