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Path space and free loop space.

1 Introduction

Let ([M.sup.d], g) be a Riemannian manifold. Fix a point 0 [member of] M. Let W(M, v) be the probability space of continuous paths in M, starting from 0 and v is Wiener measure on this space, v is defined in terms of the heat kernel on M, which is the solution to the heat equation using the Laplace Beltrami operator. I will not give further details, referring the reader to other resources for more information.

Analysis on W(M, v) is an active area of research, which began with [Dri92]. The Riemannian geometry of path space is described in detail in [CM02], whereby the Levi-Civita connection is given explicitly.

The authors in [CM02] further introduced a Markovian connection. In [Fan01], Fang computed the curvature of the Markovian connection. In [CM02] or [Fan01], they considered the space of continuous but nowhere differentiable paths supported by Wiener measure, while this article only consider smooth paths or loops.

Although path space analysis uses stochastic calculus, the calculations can be easily adapted to [P.sub.o]M, the space of [C.sup.[infinity]] based paths. This article contains 2 parts. The first part, is a very quick survey on the essential concepts in path space analysis, without the stochastic analysis.

The second part of this article, applies the analysis in path space to the free loop space LM, the space of smooth loops in M. Given [gamma] [member of] LM, [gamma] : [S.sup.1] [right arrow] M. LM is an infinite dimensional, paracompact manifold, modeled on the topological vector space [LR.sup.d], with the topology of uniform convergence of the functions and all their derivatives. (See Chapter 3 in [PS86].) A major difficulty is that there is no canonical frame along a loop, unlike the case of based paths, whereby given a fixed frame at o [member of] M, there is a horizontal lift of the path in M to a path in the principal O(M) bundle, hence defining a frame along the path.

Define a [G.sup.0] metric on LM, in Definition 3.15. The Levi-Civita connection on LM is defined by Definition 3.10 and the curvature is computed following the calculations in [Fan01]. See Theorem 3.20.

2 Analysis on Path Space

I will begin by giving a quick review of the analysis on path space in preparation for the loop space case.

2.1 Principal O(M) Bundle

Let ([M.sup.d], [[nabla].sup.M] = [nabla], g, 0, [u.sub.0]) be a smooth compact d-dimensional Riemannian manifold with the Levi-Civita covariant derivative [nabla], a Riemannian metric g, a fixed base point 0 [member of] M and a fixed orthogonal frame [u.sub.0] : [R.sup.d] [right arrow] [T.sub.o]M. In [R.sup.d], o will be the origin. Consider the principal O(M) bundle, [pi] : O(M) [right arrow] M.

Given a covariant derivative on TM, I now describe how to lift this covariant derivative on the principal O(M)-bundle. Write E [equivalent to] Hom([R.sup.d], TM), where Hom means linear transformations. E is a vector bundle over M with fiber [E.sub.m] [equivalent to] Hom([R.sup.d], [T.sub.m]M) for each m. If u is a differential curve in O(M), define [nabla]u/ds [member of] [[GAMMA].sub.m](E), by

[nabla]u/ds(s) x [xi] = [nabla](u(s)[xi])/ds (2.1)

for all [xi] [member of] [R.sup.d]. u[(s).sup.-1][nabla]u maps [R.sup.d] onto itself, i.e. it is a linear transformation on [R.sup.d]. Because the connection is Riemannian, [u.sup.-1][nabla]u/ds is in so(d), the Lie algebra of the Lie group SO(d).

Definition 2.1. (connection 1-form w) Define the connection 1-form [omega] = [[omega].sup.[nabla]] on O(M) with values in so(d) by

[omega](u'(s)) = u[(s).sup.-1] [nabla]u/ds(s)

where u(s) is any smooth path in O(M).

Definition 2.2. (canonical 1-form v) The canonical 1-form on O(M) is the 1-form v: [T.sub.u]O(M) [right arrow] [R.sub.d] given by

v([xi]) = [u.sup.-1] [[pi].sub.*][xi]

for all [xi] [member of] [T.sub.u]O(M) and u [member of] O(M).

Definition 2.3. (Horizontal vector fields) The standard horizontal vector fields B(a) [member of] [GAMMA](TO(M)) for a [member of] [R.sup.d], that is, for each a [member of] [R.sup.d], define a section of vector fields in TO(M) by the following: For each u [member of] O(M), B(a)(u) is the horizontal lift of ua [member of] TM to [T.sub.u]O(M), i.e.

B(a): O(M) [right arrow] [T.sub.u]O(M)

u [right arrow] lift of ua [member of] TM to [T.sub.u]O(M).

Note that B has 2 arguments, a [member of] [R.sup.d] and u [member of] O(M). Usually the u argument will be suppressed. Alternatively, B(a)(u) is the unique element in [T.sub.u]O(M) such that

1. [[pi].sub.*]B(fl)(u) = ua or v(B(a)(u)) = a.

2. [omega](B(a)(u)) = 0.

Denote the horizontal tangent space to u by [H.sub.u].

Let me now describe the vertical vector fields.

Definition 2.4. (Vertical vector fields) For each V [member of] so(d), define some kind of lift of this vector to a vector [??] in [T.sub.u]O(M) by

[[??].sub.u] = d/dt(u x exp(tV))|[sub.t=0]. (2.2)

This map, V [right arrow] [??] gives an isomorphism

so(d) [equivalent to] Vu

where [V.sub.u] is the vertical tangent space to u. Note that [[pi].sub.*][??] = 0 by definition.

Given a tangent vector [[xi].sub.u] [member of] [T.sub.u]O(M), I can decompose the vector into horizontal and vertical components, i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Recall that given V [member of] so(d), [??] [member of] [T.sub.u]O(M) is given by Equation (2.2).

Notation 2.5. Given X [member of] TO(M), denote the horizontal component by HX, i.e. HX = B(v([X.sub.u])) and the vertical component by VX, i.e. VX = [??].

The following formulas are stated without proof.

Proposition 2.6. 1. v(B(a)) = a (By definition).

2. [omega]([??]) = V.

3. ([[??], B(a))) = B(Va).

4. v([??]) = 0. (By definition of [??], u is fixed for all t. Hence the projection is always [pi]u for all t.)

5. [omega]([[??], [??]]) = [V,W].

Since [[pi].sub.*] is an isomorphism between [H.sub.u] and [T.sub.[pi]u]M, I can identify

(u, B(a)) [member of] [H.sub.u] [left and right arrow] ua [member of] [T.sub.[pi]u]M.

Under this identification, I will write [omega](ua) := [omega](B(a)(u)). (I am abusing the notation here, since [omega]; is a 1-form in TO(M). Here, ua [member of] [T.sub.[pi]u]M.)

Proposition 2.7. Identify [H.sub.u] with [T.sub.[pi]u]M. Alternatively, one can replace [omega] to by [omega]([[pi].sup.-1.sub.*]).

Then

[u.sup.-1]R(ua, ub)uc = (d[omega] + [omega] [conjunction] [omega]) (ua, ub)c.

The proof is omitted.

2.2 Structure Equations

Definition 2.8. (i) The curvature tensor of [nabla] is defined by

R(X,Y)Z = [[nabla].sub.X][[nabla].sub.Y]Z - [[nabla].sub.Y][[nabla].sub.X]Z - [[nabla].sub.[X,Y]]Z,

where X, Y, Z [member of] [GAMMA](TM).

(ii) The torsion tensor of [nabla] is defined by

T(X,Y) = [[nabla].sub.X]Y - [[nabla].sub.Y]X - [X,Y],

where X, Y [member of] T(TM).

(iii) The curvature form [OMEGA] of to is the so(d)-valued 2-form on O(M) defined by

[OMEGA](X,Y) = dw(HX, HY) = (dw)H(X, Y),

where X, Y [member of] TO(M) and HX and HY are the horizontal components of X and Y.

(iii') For all u [member of] O(M) and a, b [member of] [R.dup.d], set

[[OMEGA].sub.u](a, b) = [OMEGA](B(fl)(u), B(b)(u)) [member of] so(d).

(iv) The torsion form [THETA] of w is the [R.sup.d]-valued 2-form on O(M) defined by

[THETA](X, Y) = d[[theta].sup.H](X, Y) [equivalent to] d[theta](HX, HY)

for all X, Y [member of] TMO(M) and m [member of] O(M).

(iv') For all u [member of] O(M) and a, b [member of] [R.sup.d], set

[[THETA].sub.u](a, b) = [THETA](B(a)(u), B(b)(u)) [member of] [R.sup.d].

Lemma 2.9. (Structure Equations)

i [THETA] = dv + [omega] [conjunction] v. (first structure equation);

ii [OMEGA] = d[omega] + [omega] [conjunction] [omega]. (second structure equation);

iii [[OMEGA].sub.u](a, b) = [u.sup.-1]R(ua, ub)u for all u [member of] O(M) and a, b [member of] [R.sup.d].

iv [[THETA].sub.u](a, b) = [u.sup.-1]T(ua, ub) for all u [member of] O(M) and a, b [member of] [R.sup.d].

The proof is omitted and can be easily found in texts. For example see Section III, Theorem 2.4 and Section III.5 in [KN96].

2.3 Horizontal Lift

First, I will set the following convention. Derivatives with prime will denote differentiation with respect to s and dot will denote differentiation with respect to t. Note that s will be reserved for the argument of a path, i.e. o~(s). Most of the time, I will omit the argument s. In summary,

d/ds X(s, t) = X'(s, t)

d/ds X(s, t) = [??](s, t).

Definition 2.10. An absolutely continuous path a has finite energy if

[G.sup.1]([sigma]', [sigma]') := [[integral].sup.1.sub.0] g([sigma]', [sigma]') ds < [infinity].

The following notations are put together for convenience.

Notation 2.11. 1. Let H(M) be the space of absolutely continuous paths in M with finite energy, starting from o. I will reserve [sigma] for a path in M.

2. Let H(O(M)) be the space of absolutely continuous paths in O(M) with finite energy, with initial frame [u.sub.0]. I will reserve u for a path in O(M).

3. Let H([R.sup.d]) be the space of absolutely continuous paths in [R.sup.d] with finite energy, starting from the origin. I will reserve w and hfor a path in [R.sup.d]. h will denote a vector field in H(T0M) which I will of course identify with H([R.sup.d]) using [u.sub.0].

Some more definitions.

Definition 2.12. 1. A path u in H(O(M)) is said to be horizontal if [nabla]u(s)/ds = 0 or equivalently, w(u'(s)) = 0. Denote the space of absolutely continuous horizontal paths in O(M) by Hor (O(M)).

2. For a path [sigma] [member of] H([sigma]), define H([sigma]), called the horizontal lift of [sigma] [member of] H(M) to Hor(O(M)) by

H : [sigma] [right arrow] H([sigma]) = u [member of] Hor(O(M))

such that

[pi]u = [sigma].

Definition 2.13. Define a map [PHI] : H([R.sub.d]) [right arrow] Hor(O(M)) as follows: Given a path w [member of] H([R.sub.d]), define [PHI](w) = u [member of] Hor(O(M)) as the unique solution to the differential equation

u'(s) = B(w'(s))(w'(s)), u(0) = [u.sub.0]. (2.3)

The following theorem, taken from pages 281 and 282, Theorem 2.1 in [Dri92], will be stated without proof.

Theorem 2.14. The sets Hor(O(M)), H(M) and H([R.sup.d]) are in one to one correspondence. In particular, the map

[PHI] : H([R.sup.d]) [right arrow] Hor(O(M))

and the projection

[pi] : Hor(O(M)) [right arrow] H(M)

are bisections. Furthermore, the inverse of n is the horizontal lift map H and w = [[PHI].sup.-1] is given by

w(s) = [[integral].sup.s.sub.0] v(u'(r))dr.

Definition 2.15. (Carton's Development map) The map I = [pi] [omicron] [PHI] is known as the Car tan's Development map.

Now onto the geometry of the spaces. Pick a path h [member of] H([R.sup.d]). Note that one should also think of h as a vector field in H([R.sup.d]). For a [member of] H(M), define a vector field along [sigma], labelled [X.sup.h]([sigma]) by

[X.sup.h]([sigma])(s) = H([sigma])(s) x h(s), s [member of] [0,1]. (2.4)

Now define a flow of this vector field [X.sup.h](x) along a. So this flow will have 2 variables s and t. I will reserve s for the path and t for the flow. Define the flow along [X.sup.h], starting at [sigma] to be the solution [alpha] : R [right arrow] H(M) by

[??](f) = [X.sup.h]([alpha](t)) = H([alpha](t)) x H, [alpha](0) = [sigma]. (2.5)

Again, I suppress the variable s, which is the variable reserved for a path in H(M). The second condition says that [alpha](s, 0) = [sigma](s).

Remark 2.16. Such a solution [alpha] to the functional differential Equation (2.5) exists and is unique. See Remark 2.2 on page 282 in [Dri92].

The next theorem is very important, because it says how to differentiate a horizontal lift u by [X.sup.h]. Before I begin, here is another definition.

Notation 2.17. For a in H([R.sup.d]), define afunctional on H(M) by

[q.sub.a](s) = [[integral].sup.s.sub.0] [[OMEGA].sub.u](r)(a(r), w'(r))dr.

The frame u in question here is u = [u.sub.w] = [PHI](w) and I use the 1-1 correspondence, [sigma] = [pi] [omicron] [PHI](w).

Theorem 2.18. Assume all the notation as above and let [alpha](t) be a solution to the flow equation (2.5). Let u(t) = H([alpha](t)) [member of] Hor(O(M)) be a horizontal lift of [alpha](t) to the space of horizontal paths in H(O(M)). Also define

w(t) = [[PHI].sup.-1]u(t)) [member of] H([R.sup.d]).

Then u(t) and w(t) both satisfy

u(t) = [??]([??]) + B(h)(u(t)) (2.6)

and

[??]' = [q.sub.h]w' + [[THETA].sub.u](h, [w'.sub.t) + h'. (2.7)

In terms of all the parameters, Equation (2.6) is written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and Equation (2.7) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here, [u.sub.t] = u(x, t) and [w.sub.t] = w(x, t).

Remark 2.19. Only h is independent of t. u, [q.sub.h], w are all dependent on t.

Proof. Note that [??] is a tangent vector in H(O(M)). So recall I can split a vector into horizontal and vertical components as follows,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So the proof is just computing [??]and v([??](t)). Let me first compute v([??](t)). By definition, the horizontal lift of u(t) is given by u(t) = H([alpha](t)). By definition, [pi](u(t)) = [alpha](t) and thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The second equality follows because the push forward of [??](t) by [pi] is just [??](t). Now lets compute [omega]([??](t)). Pick any fixed vector a [member of] [R.sup.d], independent of s. Then by definition of [omega],

[omega]([??](t))a = u[(t).sup.-1] [nabla]/dt(t)a,

or

u(t)[omega]([??](t))a = ([??](t))a [nabla]/dt(t)a.

Take covariant derivatives on both sides and since [nabla]u(s, t)/ds = 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

But [??] = uh and by Definition 2.13 and Theorem 2.14, since u' = B(w')(u) and by taking [[pi].sub.*], I have [alpha]' = [pi].sub.*]u' = [[pi].sub.*]B(w')(u) = uw'. Hence

d/ds [omega]([??])a = [u.sup.-1] R(uw', uh)ua = [[OMEGA].sub.u](w', h)a

by Lemma 2.9. Integrate with respect to s,

[omega][??])a = [integral].sup...sub.0] [[OMEGA].sub.u](w'(r), h(r))adr = - [q.sub.h]a or

[omega]([??])a = [integral].sup.s.sub.0] [[OMEGA].sub.u](w'(r), h(r))adr = - [q.sub.h](s).

This completes the proof of Equation (2.6). To prove Equation (2.7), note that by definition of the map [PHI], u' = B(w')(u) and hence w' = v(u'). Thus to compute [??]' is the same as computing dv(u' (t))/dt. First note that u(s) is horizontal for eachs and hence [omega](u') = 0. Secondly, u(t) = H([alpha](t)) and thus [[pi].sub.*]u(t) = [??](t) = u(t)h. Therefore, v([??]) = h. Finally, since u' = B(w')(u) and hence v(u') = w'. Using structure equations, I have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3 Geometry of Loop space

To begin with, I first specialize to the based loop space and fix a frame uq at some point o [member of] M, i.e. consider only {[gamma] : [0,1] [right arrow] M, [gamma](0) = [gamma](1) = 0}. Denote this space of based loops by [L.sub.o]M. Furthermore, I will only consider oriented manifolds M. Since the calculus on based path space is so well understood, I will capitalize on it and view the space of based loops as a submanifold in based path space. The advantage of this point of view is obvious. Some of the results in based path space carry over with minor modifications. Last but not least, all the loops are considered to be smooth.

Unfortunately, there is a price to pay. When I consider based loops at o, [L.sub.o]M, I am in fact fixing a parametrization of the loop, identifying a 'starting' point and frame, denoted by [u.sub.o]. To be more precise, a parametrized loop [gamma] maps ([e.sup.2[pi]s] : s [member of] [0,1]} to M. As such, I can label the argument in [gamma] as s and s = 0 will be the starting point. At the end of the day I have to make sure that any definition made, is independent of the parametrization used.

I will now show how to locally trivialize TLM. Note that here, I am considering free loop space LM, not based loop space. Fix a loop [[gamma].sub.0] [member of] LM and pick any point on it, call it o : = [[gamma].sub.0](0), hence parametrizing the loop. I will abuse notation and write {[gamma](s) : 0 [less than or equal to] s [less than or equal to] 1, [gamma](0) = [gamma](1)} to mean a parametrized loop. Assign a local frame [u.sub.a] in an open neighborhood B in M containing [[gamma].sub.0](0) = 0. Specifically, choose a local orthonormal frame field ([f.sub.1], ..., [f.sub.d]) in the neighborhood B with the identification [f.sub.i](x) = [u.sub.0](x)[e.sub.i], x [member of] B. Here, ([e.sub.1], ..., [e.sub.d]) is the canonical orthonormal basis in [R.sub.d].

For any parametrized loop {[gamma](s) : 0 [less than or equal to] s [less than or equal to] 1, [gamma](0) = [gamma](1) [member of] B} [member of] [[union].sub.x[member of]B] [L.sub.x] M, there is a unique horizontal lift of [gamma] in H(SO(M)), that traverse along [gamma] once. Call this lift [u.sub.[gamma]](x) [equivalent to] {[u.sub.[gamma]](s) : 0 [less than or equal to] s [less than or equal to] 1), with [u.sub.[gamma]](0) = [U.sub.0]. Another description of the horizontal lift is that it is the parallel translation operator along [gamma]. As the path traverse one round, [u.sub.[gamma]](1) : [R.sup.d] [right arrow] [T.sub.o]M may not be equal to [u.sub.0].

Definition 3.1. For each {[gamma](s) : 0 [less than or equal to] s [less than or equal to] 1, [gamma](0) = [gamma](1) [member of] B} [member of] [[universal].sub.x[member of]B] [L.sub.x]M, define the holonomy operator [h.sub.[gamma]] [member of] SO(d) such that [u.sub.[gamma]](1) = [u.sub.[gamma]](0)[h.sub.[gamma]], where [u.sub.[gamma]] is the unique horizontal lift of [gamma], with an initial frame [u.sub.[gamma]] [equivalent to] [u.sub.[gamma]](0) : [R.sup.d] [right arrow] [T.sub.0]M.

Remark 3.2. This definition is dependent on the initial frame [u.sub.0] used, up to conjugacy by SO(d).

Now this holonomy operator h : [[union].sub.x[member of]B] [L.sub.x]M [right arrow] SO(d) defines a continuous smooth map on the compact-open topology. Set [g.sub.0] := [h.sub.[gamma]0] and note that [[gamma].sub.0] [member of] [L.sub.0]M. As SO(d) is a compact connected Lie group, there is a [[xi].sub.0] [member of] so(d) such that exp([[xi].sub.0]) = [g.sub.0].

Now the exponential map may not be a local diffeomorphism at [[xi].sub.0]. However, it is a local diffeomorphism at the origin. Choose an open set U containing 0 in the Lie algebra 50(d) such that the exponential map is a diffeomorphism onto an open set V = exp(U) containing the identity e of SO(d).

Consider the path [alpha] : s [member of] [0,1] [??] exp(s[[xi].sub.0]]), a path joining e to [g.sub.0]. Let [L.sub.g] denote left multiplication by g. For each [xi] [member of] U [subset or equal to] so(d), define a left invariant vector field [L.sub.g,*][xi] on SO(d). Solve the flow equation for [[beta].sub.s]([xi], x) = [B.sub.s](x),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.1)

The solution is given by

[[beta].sub.s]([xi], t) = exp(s[[xi].sub.0])exp(st[xi]), s,t [member of] [0,1]. (3.2)

Each [xi] [member of] U [subset or equal to] so(d) defines a path 0 [less than or equal to] s [less than or equal to] 1 [??] [[beta].sub.s]([xi], 1) which joins the identity e in SO(d) to each g [member of] [g.sub.0]V [??] SO(d). Here, [xi] [member of] U [subset or equal to] so(d) such that g = [g.sub.0] exp([xi]). h is a continuous map from [[union].sub.x[member of]B] to SO(d), O containing [[gamma].sub.0]. Without loss of generality, assume that O [subset or equal to] [h.sup.-1]([g.sub.0]V).

Notation 3.3. Define for each loop [gamma] [member of] 0, a skew symmetric matrix [??]([gamma]) [member of] U [subset or equal to] so(d) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. [beta] solves Equation (3.1) and is given explicitly by (3.2). This choice of [beta] is not canonical.

Write for each [gamma] in a neighborhood O [subset or equal to] [[universal].sub.x[member of]B] [L.sub.x]M,

[[eta].sub.[xi]([gamma])](s) := [[beta].sub.s][([xi]([gamma]),1).sup.-1], s [greater than or equal to] 0.

In future, I will drop the variable [gamma] and the s variable. It should be understood that [[eta].sub.[xi]] is a path.

Definition 3.4. Refer to Notation 3.3. Choose a small neighborhood B [subset or equal to] M such that an orthonormal frame field [u.sub.0] can be defined over B. Fix a loop [[gamma].sub.0] such that [[gamma].sub.0](0) [member of] B. Now define a local frame r for some neighborhood O [subset or equal to] [[universal].sub.x[member of]B] [L.sub.x]M, for 0 [less than or equal to] s [less than or equal to] 1,

[gamma] [member of] O [??] r([gamma], s) [equivalent to] [r.sup.[xi]]([gamma],s) := [u.sub.[gamma]](s)[[eta].sub.[xi]([gamma])](s). (3.3)

Note that [xi] depends on [gamma] and the initial local frame [u.sub.0] over a neighborhood B [subset or equal to] M. [u.sub.[gamma]] is the unique horizontal lift of [gamma], with an initial frame [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, I have a local trivialization of T [[universal].sub.x[member of]B] [L.sub.x]M using

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This local frame [r.sup.[xi]] is defined only in a neighborhood O containing some chosen and fixed loop [[gamma].sub.0]. It depends on the choice of [[xi].sub.0] [member of] so(d), which as a reminder, is defined by [h.sub.[gamma]0] = exp([[xi].sub.0]).

Definition 3.5. In future, drop [xi] from [r.sup.[xi]] and write it as r = u[[eta].sub.[xi]. Define [PSI]([gamma]) = (H[[eta].sub.[xi])([gamma]). Recall H([gamma]) is the horizontal lift of [gamma], with starting frame [u.sub.0]. Thus, [PSI] maps a loop in [L.sub.0]M to a loop in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] using a local trivialization. Of course, this map depends on the choice of [xi].

Definition 3.6. Let [rho](x,t) be a family of loops in the free loop space LM such that [rho](*,0) = [gamma](x) and [partial derivative][rho](x,t)/[[partial derivative]t|.sub.t=0] = v(x), v(x) [member of] [T.sub.[gamma]]LM. Define for a smooth function F : LM [right arrow] S, S a manifold,

[D.sub.v]F([gamma])] := [partial derivative]F([rho](*,t))/[partial derivative]t|[sub.t=0] [member of] [T.sub.F([gamma])]S.

Now suppose I fix a point 0 [member of] M and a frame [u.sub.0] at 0. Given a based loop [gamma] [member of] [L.sub.0]M and a tangent field v [member of] [T.sub.[gamma]][L.sub.0]M, I want to compute [D.sub.v]r [member of] TLSO(M), and for each 0 [less than or equal to] s [less than or equal to] 1, [D.sub.v]r(s) [member of] TSO(M) has horizontal and vertical components.

Definition 3.7. Lei t? be a loop in H([R.sup.d]) and v = rv, r as defined in Definition 3.4. Define a skew symmetric matrix path A, as

A(v)(x) [equivalent to] [A.sub.v](x) = [beta]([xi]([gamma]),1)[D.sub.v][[beta].sup.-1]([xi]([gamma]),1).

Do note that [A.sub.v](x) [equivalent to] [A.sub.v]([gamma], x) is dependent on [gamma] for [gamma] [member of] O [subset or equal to] LM, since [xi] is dependent on [gamma].

The vertical vector field is given by [r.sup.-1][[DELTA].sup.M.sub.v]r(x) (See Equation (2.1).) and using Equation (2.6), is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The horizontal vector field, however is the unique vector field such that

rv([D.sub.v]r) = [pi]*[D.sub.v]r = rv,

and thus v([D.sub.v]r) = v. Let me summarize it as an analog of Theorem 2.18.

Corollary 3.8. Fix a point o [member of] M and a frame [u.sub.0] at o and consider the based loop space [L.sub.0]M. Let [gamma] [member of] [L.sub.0]M and v([gamma], x) [member of] [T.sub.[gamma]][L.sub.0]M with v([gamma], x) = (rv)([gamma], x) = ([u[eta].sub.[xi]v])([gamma], x), v([gamma],s) [member of] [LK.sup.d] for 0 [less than or equal to] s [less than or equal to] 1 with v(0) = 0. Refer to Definition 3 A for the definition of r. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3.1 Covariant Derivative on Free Loop Space

I am now ready to do some analysis on free loop space. The obvious difference from the analysis on based loop space is that the starting point [gamma](0) is allowed to vary. Although not stated explicitly, in the case of based paths (loops), the vector fields along the path (loop) X are such that X(0) = 0. When considering free path (loop) space, there is no such restriction.

In the case of based paths (loops), the starting frame [u.sub.0] at [gamma] (0) is fixed. On free paths (loops), the starting frame [u.sub.0] is allowed to vary. This does not change much of the analysis done earlier on, except that now I have to include the derivative of [u.sub.0] using the Levi-Civita connection in the earlier computations.

Fix a [[gamma].sub.0] [member of] LM. Recall I fix a point 0 = [[gamma].sub.0](0) and choose a local frame [u.sub.0] over some open neighborhood 0 [member of] U [subset or equal to] M. Then, there exists an open neighborhood [[gamma].sub.0] [member of] O [subset or equal to] [[universal].sub.x[member of]M] [L.sub.x]M, such that there is a local trivialization of T [[universal].sub.x[member of]M] [L.sub.x]M using

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

u(x) [equivalent to] [u.sub.[gamma]](x) is the unique horizontal lift of [gamma], with initial frame [u.sub.0]([gamma](0)) = [u.sub.[gamma]](0) and [gamma](0) [member of] U. Refer to Equation (3.3) in Definition 3.4.

Definition 3.9. Let [[gamma].sub.0] [member of] O for some open neighborhood O [subset or equal to] [[universal].sub.x[member of]u] [L.sub.x]M and suppose {r(s) = u(s)[[eta].sub.[xi]](s) : 0 [less than or equal to] s [less than or equal to] 1) is a local trivialization frame over O with [u.sub.0] a local orthonormal frame field over U [subset or equal to] M. Let (z(s) : 0 [less than or equal to] s [less than or equal to] 1, z(0) = z(1)) [member of] [LR.sup.d] and Z(x) := (rz)(x). Define a connection 1-form to for 0 [less than or equal to] s [less than or equal to] 1 and [gamma] [member of] O,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.4)

Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the connection 1-form in the direction [u.sub.0]z(0). Simply put, it is the connection 1-form for the base point, at time 0. Some results done earlier transfer over to free paths (loops) by adding this additional term [lambda]. For example, [[eta].sup.-1.sub.[xi]] [[lambda].sub.v][[eta].sub.[xi]] should be added to the vertical component of the tangent field in Corollary 3.8, i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As a result, the connection 1-form [[omega].sub.z](s) = [r.sup.-1](s)[[nabla].sup.M.sub.Z(s)]r(s), the vertical component of ([D.sub.z]r)(s), for 0 [less than or equal to] s [less than or equal to] 1. Here, [[nabla].sup.M.sub.Z(s)]r(s) is given by Equation (2.1).

Continue the above set up, define a covariant derivative on free loop space as follows.

Definition 3.10. (Covariant derivative [[nabla].sup.LM] on free loop space) Let [Z.sub.i], i = 1,2 be smooth sections of TLM. Define a local trivialization over O [subset or equal to] [[universal].sub.x[member of]u] [L.sub.x]M, {r([gamma], s) : 0 [less than or equal to] s [less than or equal to] 1) as in Definition 3.4. Thus the vector fields [Z.sub.1] can be written as {[Z.sub.i]([gamma],s) : r([gamma],s)[z.sub.i]([gamma],s) : 0 [less than or equal to] s [less than or equal to] 1) for [gamma] [member of] O. Here, [Z.sub.i] : ([gamma],s) [member of] O x [0,1] [??] [z.sub.i]([gamma],s) [member of] [T.sub.[gamma](s)]M is smooth and [z.sub.i]([gamma], 0) = [z.sub.i]([gamma], 1) for any [gamma] [member of] O.

Define the covariant derivative [[nabla].sup.LM] as, for each 0 [less than or equal to] s [less than or equal to] 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

whereby [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is given by Equation (3.4). To clarify,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that ([alpha](t, x) : -[epsilon] [less than or equal to] t [less than or equal to] [epsilon], [epsilon] > 0) is any family of loops in O with [partial derivative][alpha](t,s)/[partial derivative]t[|.sub.t=0] = [Z.sub.1]([gamma],s) for 0 [less than or equal to] s [less than or equal to] 1.

The covariant derivative is defined, based on the choice of r, which fixes a starting point [gamma](0) for [gamma], hence parametrizing the field [Z.sub.i], i = 1,2. The next lemma says that the covariant derivative is well-defined.

Lemma 3.11. [[nabla].sup.LM] is well-defined, independent of the frame r used.

Proof. Let O be open in [[universal].sub.x[member of]xu] [L.sub.x]M and {r([gamma],s) = [u.sub.[gamma]](s)[[eta].sub.[xi]](s) : 0 [less than or equal to] s [less than or equal to] 1} be a frame over any [gamma] [member of] O, with a starting frame [u.sub.[gamma]](0) at [pi]([u.sub.[gamma]](0)) [member of] M, [pi] : SO(M) [right arrow] M. Let 0 [less than or equal to] [tau] [less than or equal to] 1 and suppose a new frame [??]([gamma],s) = [[??].sub.[xi]](s)[[eta].sub.[xi]](s) over [gamma] [member of] [??] [subset or equal to] [[universal].sub.x[member of]u] [L.sub.x]M is chosen, such that at s = 0,1 begin with a frame [[??].sub.[gamma]](0) at [gamma] ([[??].sub.[gamma]](0)) [member of] M. Without loss of generality, by taking the intersection of O and [??] if necessary, I will assume O = [??]. The frame f can be written for [gamma] [member of] O and 0 [less than or equal to] s [less than or equal to] 1,

[??]([gamma],s) = [u.sub.[gamma]](s + [tau])[[eta].sub.[xi]([gamma])](s + [tau])[rho]([gamma],s) = r([gamma],s + [tau])[rho]([gamma],s), p(7,s) [member of] SO(d).

Let [Z.sub.i], i = 1,2 be 2 smooth sections in TLM. For 0 [less than or equal to] s [less than or equal to] 1, in terms of the frame r, [Z.sub.i]([gamma],s) = r([gamma],s)[z.sub.i]([gamma],s) for [gamma] [member of] O, but in terms of [??],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Write [Z.sub.i,[tau]]([gamma],s) = [z.sub.i]([gamma],s + [tau]), [r.sub.[tau]]([gamma],s) = r([gamma],s + [tau]) for 0 [less than or equal to] s [less than or equal to] 1. According to Definition 3.10, for 0 [less than or equal to] s [less than or equal to] 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This quantity [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] appears often in the analysis such that it deserves a separate symbol. This quantity appears because of the derivative of the holonomy operator, which happens only for loop space.

Notation 3.12. Let O [subset or equal to] [[universal].sub.x[member of]u] [L.sub.x]M as in Definition 3.4. Recall r = [u[[eta].sub.[xi]] and write [[GAMMA].sub.u] := [[OMEGA].sub.u]([[eta].sub.[xi]]*,[[eta].sub.[xi]]*). I will write the vertical component as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4).

3.2 Lie Bracket

To compute the Lie Bracket amounts to compute the Lie Bracket over [[universal].sub.x[member of]u] [L.sub.x]M U [subset or equal to] M. Let X, Y [member of] TLM.

The reference is taken from Pages 148 to 150 in [CM02]. Throughout this subsection, fix a [[gamma].sub.0] and a starting point 0 = [[gamma].sub.0](0). Choose an open neighborhood 0 [member of] U [subset or equal to] M, fix an orthonormal frame [u.sub.0] over U [subset or equal to] M and hence define a neighborhood O [subset or equal to] [[universal].sub.x[member of]u] [L.sub.x]M and a frame r = [u[eta].sub.[xi]] for [gamma] [member of] O, as in Definition 3.4.

Suppose X = rx and Y = ry. Assume that x and y, tangent fields in H([R.sup.d]), are independent of [gamma]. Since I have a choice of the starting frame [u.sub.0], assume that [[lambda].sub.x] = [[lambda].sub.y] = 0.

Definition 3.13. (Cylinder functions) A function f is a cylinder function on H(M) if f([sigma]) = [??]([sigma]([s.sub.1]),[sigma]([s.sub.2]), ..., [sigma]([s.sub.k])), where [??] : M x ... x M [??] R. Now by projection [pi], consider f as a cylinder function on H(O(M))

f(u) = [??]([pi]u([s.sub.1]),[pi]u([s.sub.2]), ... [pi]u([s.sub.k])).

Thus f(u.) is a function from H([R.sup.d]) [right arrow] H(O(M)). Therefore, only consider in general, cylinder functions f on H(O(M)).

For a cylinder function on [[universal].sub.x[member of]u] [L.sub.x]M, f([gamma]) = F([gamma]([s.sub.1]), ..., [gamma]([s.sub.n])), lift it to a function on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i.e.

(f [omicron] [pi])([r.sub.[gamma]]) = (F [omicron] [pi])([r.sub.[gamma]]([s.sub.1]), ..., [r.sub.[gamma]]([s.sub.n])) := [??]([r.sub.[gamma]]([s.sub.1]), ..., [r.sub.[gamma]]([s.sub.n])).

Note that [??] is a function on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since cylinder functions are dense, it suffices to compute the Lie bracket on cylinder functions.

Recall [PSI] = H[[eta].sub.[xi]], H is horizontal lift. Since F = F [omicron] [pi] [omicron] [PSI] = [??] [omicron] [PSI], using Corollary 3.8,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where I adapt Einstein's summation notation and [B.sup.k.sub.[alpha]] = B([e.sub.[alpha]])(r([s.sub.k])) is the canonical horizontal vector field at r([s.sub.k]) = [u[eta].sub.[xi]]([s.sub.k]). Note that there is no derivation by a vertical tangent vector since the function is an honest cylinder function on H(M). However, note that the derivation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a cylinder function on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now differentiate a second time, using [D.sub.[gamma]], again using Corollary 3.8. Note that I have to differentiate using the vertical tangent vector,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.5)

If j [not equal to] k, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So only consider k = j. Since [B.sub.[alpha]] and [B.sub.[beta]] are horizontal vector fields, [[B.sub.[alpha]], [B.sub.[beta]]] is vertical. To show this, compute [[pi].sub.*] [[B.sub.[alpha]], [B.sub.[beta]]] = [[[pi].sub.*][B.sub.[alpha], [[pi].sub.*][B.sub.[beta]]]. This follows from the Torsion free of Levi-Civita covariant derivative on M, [[nabla].sup.M.sub.X]Y - [[nabla].sup.M.sub.Y]X = [X, Y]. As horizontal vector fields, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence [[[pi].sub.*][B.sub.[alpha]], [[pi].sub.*][B.sub.[beta]]] = 0. But [??] only depends on [pi](u), hence this term vanishes.

The first sum in Equation (3.5) vanishes. For the last sum, same reasoning: if j [not equal to] k then the fields commute. So assume j = k. Use one of the useful formulas, [[??], B(a)] = B(Va) for a skew symmetric matrix V and a vector a [member of] [R.sup.d]. Apply this formula, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, the second sum in Equation (3.5) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

So if I plug into Equation (3.5),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since F is a cylinder function on H(M), I have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus the following result.

Theorem 3.14. The Lie bracket of 2 tangent fields, X and Y, is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

More generally, the Lie bracket of X = rx and Y = ry is given by, under r,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 3.15. (Metric on LM.) Define a [G.sup.0] metric on TLM by [G.sup.0](X,Y) = [[integral].sup.1.sub.0](X(s),Y(s))ds.

With this definition of the metric, we have the following corollary.

Corollary 3.16. The connection [[nabla].sup.LM] is the Levi-Civita connection.

3.3 Curvature

Throughout this subsection, fix a [[gamma].sub.0] [member of] LM and a starting point 0 = [[gamma].sub.0](0). Choose an open neighborhood o [member of] U [subset or equal to] M, fix an orthonormal frame [u.sub.0] over U [subset or equal to] M and hence define a neighborhood O [subset or equal to] [[universal].sub.x[member of]u] [L.sub.x]M and a frame r = u[[eta].sub.[xi]] dependent on each [gamma] [member of] O, as in Definition 3.4. The goal in this subsection is to compute the curvature at the loop [[gamma].sub.0].

The calculations in this subsection follow those in Section 3 of [FanOl]. Now [gamma]' is a tangent field along the loop [gamma], so I can write for each [gamma] [member of] O [subset or equal to] [[universal].sub.x[member of]u] [L.sub.x]M, [gamma]' = r([gamma])[mu]' for some loop vector field [mu]' [member of] [LR.sup.d]. Define a map b' : [gamma] [member of] O [subset or equal to][[universal].sub.x[member of]u] [L.sub.x]M [??] [mu]' [member of] LRd. To find [D.sub.v]b' requires going back to the development map, where v [member of] TLM.

The development map I maps H([R.sup.d]) to H(M) and is a diffeomorphism. So, write I(w) = [gamma] [member of] O [subset or equal to] [[universal].sub.x[member of]u] [L.sub.x]M. Note that may not be a loop. Their relationship is

[gamma]'(s) = [u.sub.[gamma]](s)w'(s).

The anti-development map is given by

[I.sup.-1]([gamma]) = [[integral].sup.[??].sub.0] [u.sup.-1.sub.[gamma]][gamma]'dr.

Differentiate,

([I.sup.-1])'([gamma])=[u.sup.-1.sub.[gamma]][gamma]' = [[eta].sub.[xi][gamma]]u',

or by definition of b',

([I.sup.-1)' = [[eta].sub.[xi]]b'.

so, b' = [[eta].sup.-1.sub.[xi]]([I.sup.-1])'.

Lemma 3.17. Continue with the notations as above. Assume the connection on M is torsion free. Then

[D.sub.v]b' = v' + [[eta].sup.-1.sub.[xi]][[eta]'.sub.[xi]]v - [[LAMBDA].sub.v]b', (3.6)

where v = rv with v [member of] [LR.sup.d].

Proof. Torsion free implies that [THETA] [equivalent to] 0. From Equation (2.7), let [psi]' = [q.sub.h]w' + h' and write [D.sub.q,h] as the tangent vector associated to [psi], i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any smooth function on H([R.sup.d]). Recall that I = n o [PHI]. Hence by Theorem 2.18,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

So ([I.sup.-1])' o I :w [right arrow] w'. Therefore, by definition,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Recall I fix a loop [[gamma].sub.0] [member of] O. Choose an orthonormal frame field [{[e.sub.a]}.sub.a] over U [subset or equal to] M such that at the base point [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence [[lambda].sub.h] [equivalent to] 0, h [member of] [LR.sup.d] in Definition 3.9, throughout this subsection.

Lemma 3.18. Suppose for [gamma] [member of] O, [gamma]' = (rb')[gamma]), r = [u[eta].sub.[xi]] as in Definition 3.4. Write v = rv and [mu] = rw. Given v(x),w(x) [member of] L([R.sup.d]) independent of [gamma] [member of] O,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, I have to compute [D.sub.v][[OMEGA].sub.r]. Now [[OMEGA].sub.r] = [r.sup.-1]R(r,r)r and r is a frame in O(M). Using the push forward of [PSI] : [L.sub.u]M [right arrow] LO(M), [[PSI].sub.*][D.sub.v] is given by [D.sub.v]r. Thus, by Corollary 3.8,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, using Equation (3.6).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

But note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This then simplify the above expression to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Recall the Lie bracket of [D.sub.[mu]] and [D.sub.v] is given by, in term of r,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Lemma 3.19.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 3.20. Let [[gamma].sub.0] [member of] O [subset or equal to] [[universal].sub.x[member of]u] [L.sub.x]M and define r = [u[eta].sub.[xi]] as in Definition 3.4. Let v,w,z [member of] [LR.sup.d]. The curvature [R.sup.LM] of [[nabla].sup.LM] at [[gamma].sub.0] is given by

[R.sup.LM](rv,rw)rz = [R.sup.M](rv(x),rw(x))rz(x),

whereby [R.sup.M] is the curvature of the underlying manifold M.

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence this simplifies to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By the second Bianchi identity,

[[partial derivative].sub.B(v)][[OMEGA].sub.r](w,b') - [[partial derivative].sub.B(w)][[OMEGA].sub.r](v,b') = [[partial derivative].sub.B(b')][[OMEGA].sub.r](w,v)

and by the symmetry of curvature tensor, at [[gamma].sub.0] [member of] O.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now I have to compute r'. Note that by definition, r = u[[eta].sub.[xi]] and [pi]([u[eta].sub.[xi]]) = [gamma]. Thus I seek to find a horizontal vector field B such that [[gamma].sub.*]B = [gamma]'. But [gamma]' = u[[eta].sub.[xi]]b' = rb'.

Therefore [[pi].sub.*]B = rb' which implies v(r') = b'. The vertical vector field is clearly -[??]. Thus,

r' = [(b').sup.[alpha]][B.sub.[alpha]] - [??].

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

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[CM96] Ana-Bela Cruzeiro and Paul Malliavin, Renormalized differential geometry on path space: structural equation, curvature, J. Funct. Anal. 139 (1996), no. 1,119-181. MR MR1399688 (97h:58175)

[CM02] A. B. Cruzeiro and P. Malliavin, Riemannian geometry on the path space, Stochastic partial differential equations and applications (Trento, 2002), Lecture Notes in Pure and Appl. Math., vol. 227, Dekker, New York, 2002, pp. 133-165. MR MR1919507 (2003L60095)

[Dri92] Bruce K. Driver, A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold, J. Funct. Anal. 110 (1992), no. 2, 272-376. MR MR1194990 (94a:58214)

[Dri94] --, A Cameron-Martin type quasi-invariance theorem for pinned Brownian motion on a compact Riemannian manifold, Trans. Amer. Math. Soc. 342 (1994), no. 1, 375-395. MR MR1154540 (94e:60072)

[Dri99] B. K. Driver, The Lie bracket of adapted vector fields on Wiener spaces, Appl. Math. Optim. 39 (1999), no. 2,179-210. MR MR1665672 (2000b:58063)

[Fan01] Shizan Fang, Markovian connection, curvature and WeitzenbOck formula on Riemannian path spaces, J. Funct. Anal. 181 (2001), no. 2, 476-507. MR MR1821704 (2002a:58043)

[FM93] Shizan Fang and Paul Malliavin, Stochastic analysis on the path space of a Riemannian manifold. I. Markovian stochastic calculus, J. Funct. Anal. 118 (1993), no. 1, 249-274. MR MR1245604 (94i:58209)

[Hsu02] Elton P. Hsu, Stochastic analysis on manifolds, Graduate Studies in Mathematics, vol. 38, American Mathematical Society, Providence, RI, 2002. MR MR1882015 (2003c:58026)

[KN96] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. I, Wiley Classics Library, John Wiley & Sons Inc., New York, 1996, Reprint of the 1963 original, A Wiley-Interscience Publication. MR MR1393940 (97c:53001a)

[PS86] Andrew Pressley and Graeme Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1986, Oxford Science Publications. MR MR900587 (88i:22049)

[Str00] Daniel W. Stroock, An introduction to the analysis of paths on a Riemannian manifold, Mathematical Surveys and Monographs, vol. 74, American Mathematical Society, Providence, RI, 2000. MR MR1715265 (2001m:60187)

Received by the editors February 2010--In revised form in March 2010.

Communicated by S. Gutt. 2000 Mathematics Subject Classification : 58D15.

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