ALEXEY V. GAVRILOV. I p (v) : T p M T Expp vm the operator of parallel transport along the geodesic γ v. Let E p be the map

Transcription

1 THE TAYLOR SERIES RELATE TO THE IFFERENTIAL OF THE EXPONENTIAL MAP arxiv: v [math.g] 3 May 202 ALEXEY V. GAVRILOV Abstract. In this paper we study the Taylor series of an operator-valued function related to the differential of the exponential map. For a smooth manifold M with a torsion-free affine connection the operator E p(v) acting on the space T pm is defined to be the composition of the differential of the exponential map at v T pm with parallel transport to p along the geodesic. The Taylor series of E p as a function of v is found explicitly in terms of the curvature tensor and its high order covariant derivatives at p. Key words: affine connection, exponential map, Taylor series. 200 Mathematical Subject Classification: 53B05.. INTROUCTION Let (M, ) be a smooth manifold with a torsion-free affine connection. For a point p M, consider the exponential map Exp p : T p M M. Let v T p M be a vector at a fixed point p and γ v be the corresponding geodesic γ v (t) = Exp p (tv). As T p M is a linear space, the differential of the exponential map may be considered a linear map of the form enote by d v Exp p : T p M T Expp vm. I p (v) : T p M T Expp vm the operator of parallel transport along the geodesic γ v. Let E p be the map defined by E p : T p M End R (T p M), E p (v) = I p (v) d v Exp p. This is a smooth map between two linear spaces, hence its Taylor series is well defined. Our goal is to find this series explicitly in terms of the curvature tensor. For the best of author s knowledge, the problem has never been considered in this generality. However, for a symmetric space M the answer is well known [2] (see also [4], Ch. IV, Theorem 4.). In our terms E p (v) = k=0 (2k +)! rk 0 (v); ()

2 2 ALEXEY V. GAVRILOV the operator r 0 is defined by r 0 (v) : w R p (v,w)v, w T p M, whereristhecurvaturetensor. Notethatthisisaveryspecialcase, because for a more general manifold the series depends not only on the curvature tensor itself, but on its high order covariant derivatives at the point p as well. Apparently, the only relevant result known for a general affine (or Riemannian) manifold is the Helgason s formula [3] (in fact, the formula is proved for an analytic manifold with an affine connection). The formula is d v Exp p w = e adv adv w Expp v, where the adjoint refers to the Lie algebra of smooth vector fields and v denotes the vector field defined by the condition v Exp p u = I p (u)v, u T p M (the same for w ). This, of course, is not a Taylor series, and its relation to the problem remains quite obscure. The author might also mention his own paper [], where an algorithm of computing the Taylor series of the inverse operator E p (v) (denoted there by H(v)) is proposed. This algorithm, however, is quite involved in comparison with the explicit formula presented here. Remark. In one respect the Helgason s formula is more general than the results obtained below: it is not necessary to assume the connection to be torsion-free. It should be noted, however, that the differential of the exponential map (as the map itself) does not depend on the torsion part of the affine connection. So, it is natural to restrict ourselves to the torsion-free case. In a more general case the Taylor series contains terms which depend on the torsion tensor, but these terms actually have nothing to do with the exponential map. 2. THE SERIES Before we can formulate the main result we have to introduce some notation. For n 0 and v T p M denote by R p,v (n) the n-th order covariant derivation of the curvature tensor at the point p in the direction v. That is, R (n) p,v = v n ( n R) p. (The common notation n v,...,v for hight order covariant derivation in the direction v is not very convenient when n is variable. For this reason we use the contraction notation. The sign on the right hand side denotes the contraction of the polyvector v n with the tensor n R). For p,v,n as above denote by r p,n (v) End R (T p M)

3 IFFERENTIAL OF THE EXPONENTIAL MAP 3 the linear operator defined by r p,n (v) : w R (n) p,v (v,w)v, w T pm. For the sake of convenience we usually omit the point and, sometimes, the vector: For example, r n = r n (v) = r p,n (v). r 0 : w R p (v,w)v, r : w ( v R) p (v,w)v, etc. Obviously, r n as a function of v is homogeneous of degree n+2 r n (tv) = t n+2 r n (v), t R. We also need an appropriate notation for compositions of operators of this kind. Call a finite sequence of nonnegative integers a list. The set of all lists (including the empty one) is denoted by Λ. The empty list is denoted by the symbol ; to write down a nonempty list we use square brackets; for example, ν = [2,0,] Λ. For every list ν Λ there is a corresponding operator r ν End R (T p M) whichisacomposition ofsimpleoperators r n. Namely, ifν = [n,n 2,...,n k ] then r ν = r n r n2...r nk. By definition, r = ½. We shall need three number functions on lists: the factorial ν!, the degree ν and the denominator c ν. By definition,! =, = 0. For ν = [n,n 2,...,n k ] we define the factorial and the degree as follows Obviously, ν! = k (n j!), ν = 2k+ j= k n j. j= r ν (tv) = t ν r ν (v), t R, which is where the term degree comes from. The denominator is defined by c = and a recurrent relation c ν = ν ( ν +)c ν, ν Λ\{ }, where ν is obtained from ν by omitting the first element from the list. For example, [2,0,] = = 9, hence c [20] = 9 0c [0] = 90 30c [] = c = Now we are able to formulate the main result.

4 4 ALEXEY V. GAVRILOV Theorem Let (M, ) be a smooth manifold with a torsion-free affine connection. Let p M and v T p M. Then for any n 0 the following equality holds d n n! dt ne p(tv) = r ν (v), (2) ν!c ν ν =n where the sum on the right hand side is taken over all the lists ν Λ of degree n. In other words, we have the Taylor series E p (v) = r ν (v). (3). ν!c ν ν Λ Forexample, thereare3listsofdegreenotgreaterthan6, namely,[0],[],[2], [0,0],[3],[,0],[0,],[4],[2,0],[,],[0,2],[0,0,0]. Computingthecorresponding coefficients, we have the Corollary The function E p can be expressed as follows E p (v) = ½+ 6 r 0+ 2 r + 40 r r r r r r 0r r r 2r r r 0r r3 0 +ρ 7(v), where ρ 7 (v) = O( v 7 ) as v 0. Note that for a list ν = [0,0,...,0] which consists of k zeros we have ν = 2k, hence c ν = 2k(2k +)c ν = (2k +)!. Thus, the coefficient at the term r ν = r0 k in the Taylor series is equal to /(2k +)!, in agreement with (). It may be shown that there are no nontrivial algebraic relations between the operators r n on a general manifold. So, the series (3) is unique as a formal series in non-commutative variables r n. 3. PARALLEL TRANSPORT OF THE CURVATURE TENSOR We shall need some properties of the covariant derivation along a geodesic. For our purpose it is convenient to consider a connection on a vector bundle instead of an affine connection(which is a connection on the tangent bundle). Let E M be a smooth vector bundle on a smooth manifold. The space of smooth sections of E is denoted by C (M,E). The bundle is provided with a connection, which is a linear map satisfying the Leibniz rule : C (M,E) C (M,T M E), fu = df u+f u, f C (M),u C (M,E). Note that the manifold itself is not supposed to have a connection for now. Let γ : I M be a smooth curve, where I R is an interval. The induced connection on the restricted bundle γ E I is usually denoted

5 IFFERENTIAL OF THE EXPONENTIAL MAP 5 by the sign. It is convenient to use this connection in the form of a differential operator dt, where t I is the parameter on the curve. This operator is called a covariant derivation along the curve γ. In other words, for any section u C (M,E) we have the equality where dt γ u = γ γ u, (4) γ(t) = d dt γ(t) T γ(t)m, t I. To be less pedantic, one may write (4) as an operator identity dt = γ. We shall need the following simple, but useful Lemma Let (M, ) be a smooth manifold with an affine connection and E M be a smooth vector bundle with a connection. Let γ : I M be a geodesic. Then for any n and any smooth section u of the bundle E n dt nγ u = γ n γ n u, (5) where dt is the covariant derivation along γ. The equality (5) may also be written in the operator form n dt n = γ n n. Apparently, this formula is well known, but the author does not know a proper reference. For this reason, we present here a proof. Note that for n > the section n u on the right hand side of (5) depends on the affine connection on the manifold while the left hand side depends on the curve γ only. Proof. It is an almost trivial fact that the derivation dt can becanonically extended to the products of E with tensor bundleson M and inherits all the common properties of the covariant derivation. In particular, it is compatible with contraction. For n = the formula (5) is the definition of dt. If it is valid for some n, then we have n+ n dt n+γ u = dt dt nγ u = dt γ n γ n u = By assumption, γ is a geodesic, hence ( ) dt γ n γ n u+ γ n dt γ n u. dt γ = 0 and the first term on the right hand side vanishes. Applying (4) to the section n u C (M,T M n E), we have dt γ n u = γ γ n+ u.

6 6 ALEXEY V. GAVRILOV Thus, n+ dt n+γ u = γ n+ γ n+ u, and the equality (5) follows by induction. The covariant derivation is closely related to parallel transport. As above, denote by I p = I p,e the operator of parallel transport of the form I p (v) : E p E Expp v along the geodesic γ v (t) = Exp p (tv). By the definition of parallel transport, dt I p(tv)z = 0, where dt is the covariant derivation along γ v and z E p is a constant. In a more general case, when z = z(t) depends on the parameter t, we have the operator equality dt I p(tv) = I p (tv) d dt, (6) which will be of use below. After the above preliminaries we have come to the matter. Consider the operator R p (v) End R (T p M), defined by R p (v)w = I p (v) R Expp v(i p (v)v,i p (v)w)i p (v)v,w T p M, (7) where R x denotes the curvature tensor at a point x M. As is well known, parallel transport is compatible with tensor operations. Thus, this operator can also be written in the form R p (v)w = (I p (v) R Expp v)(v,w)v. In the latter equality the parallel transport operator I p (v) is applied to the curvature tensor, i.e. it acts on the bundle T3 M. Lemma 2 For v T p M and n 0, d n dt nr p(tv) = n(n )r n 2 (v). (8) In fact, r and r 2 are not defined, but it is convenient to consider them arbitrary operators. So, for n = 0 and n = the right hand side of (8) is zero. Proof. Consider the geodesic γ = γ v. For w T p M we have hence By (6), R p (tv)w = t 2 (I p (tv) R γ(t) )(v,w)v, d n ( dt nr p(tv) d n 2 w = n(n ) dt n 2I p(tv) R γ(t)) (v, w)v. d n 2 dt n 2I p(tv) R γ(t) = I p (tv) n 2 dt n 2R γ(t).

7 By Lemma, IFFERENTIAL OF THE EXPONENTIAL MAP 7 n 2 dt n 2R γ(t) = v n 2 ( n 2 R ) p = R(n 2) p,v. Taking into account the equality I p (0) = ½ and the definition of r n 2, we have (8). 4. THE JACOBI FIEL For given vectors v,w T p M consider a family of geodesics γ s (t) = Exp p (tv +tsw), parametrized by a real number s taken from a neighbourhood of zero. enote by γ = γ 0 = γ v the geodesic corresponding to s = 0. Let J C (I,γ TM) be the vector field defined by J t = d ds γ s(t) T γ(t) M, s=0 which is equivalent to J t = d tv Exp p tw. It is well known that a field of this kind is a Jacobi field [5]. This means that it satisfies the differential equation 2 dt 2J t = R γ(t) ( γ(t),j t ) γ(t). (9) Lemma 3 For any v T p M the operator E p (tv) satisfies the differential equation Proof. By the definition of E p, (t 2 d2 dt 2 +2t d ) E p (tv) = R p (tv)e p (tv). (0) dt E p (tv)tw = I p (tv) d tv Exp p tw = I p (tv) J t. Thus, by (6) and (9), (t d2 dt 2 +2 d )E p (tv)w = d2 dt dt 2E p(tv)tw = I p (tv) 2 dt 2J t = I p (tv) R γ(t) ( γ(t),j t ) γ(t). On the other hand, by the definition of the operator R, we have the equality R p (tv)e p (tv)w = ti p (tv) R γ(t) ( γ(t),j t ) γ(t). Comparing these equalities and taking into account that they are valid for any choice of w, we have (0). Proof of Theorem. For n 0 denote the left hand side of (2) by E n = d n n! dt ne p(tv).

8 8 ALEXEY V. GAVRILOV The equality E n = ν =n for n = 0 and n = can be verified directly: E 0 = r = ½, E = 0. ν!c ν r ν () Taking the n-th order derivative of the both sides of (0) at t = 0, we have by Lemma 2 the equality n ( ) n n(n+)n!e n = k(k )r k 2 (n k)!e n k. k k=0 For n 2 the equality takes the form n 2 E n = n(n+) m! r me n m 2. (2) m=0 By induction, we may assumethat the formula () is valid for the operators E n m 2 on the right hand side of (2). We have then the equality E n = n 2 m=0 µ =n m 2 m!µ!n(n+)c µ r m r µ. enote ν = [m,µ] (then ν = µ). One can see that the double sum in the latter equality can be replaced by a single sum taken over the lists ν of degree n. Taking into account the obvious equalities we obtain (). ν! = m!µ!, c ν = n(n+)c µ, r ν = r m r µ, References [] A.V.GAVRILOV, The Leibniz formula for the covariant derivative and some applications(in Russian), Mat. Tr. (Matematicheskie Trudy)3(200), [2] S. HELGASON, On Riemannian curvature of homogeneous spaces, Proc. Amer. Math. Soc. 9(958), [3] S. HELGASON, Some remarks on the exponential mapping for an affine connection, Math. Scand. 9(96), [4] S. HELGASON, ifferential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics. 34, AMS, Providence, Rhode Island, 978. [5] S. KOBAYASHI, K. NOMIZU, Foundations of ifferential Geometry, II, Interscience Publisher, New York, 963. Alexey V. Gavrilov, epartment of Physics, Novosibirsk State University, 2 Pirogov Street, Novosibirsk, , Russia gavrilov9@gmail.com