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Determination of the metric from the connection.

1. Introduction

One of the fundamental principles of differential geometry is that a Riemannian manifold (M, h) uniquely determines a connection: the Levi-Civita connection of the metric h. We are interested in the inverse problem of obtaining the parallel metrics of the Levi-Civita connection for a given Riemannian metric. This corresponds to finding the general solution to a system of global PDEs defined on M. We demonstrate that genetically these solutions can be explicitly constructed by an algebraic procedure.

This problem is closely tied to the concept of the holonomy group. Indeed, knowledge of the invariant decomposition of the tangent space at a point of the manifold, with respect to the holonomy group, yields the set of parallel metrics. However, the determination of the action of the holonomy group is an intractable problem in general, as it relies upon solutions to differential equations defined on the manifold. It is shown below that the question for generic connections is, in fact, a combinatorial/algebraic one that does not require such analytical information; the invariant decomposition of the tangent bundle of the manifold can be obtained by solely algebraic means. Our method, therefore, is constructive and not merely existential.

The following section begins by introducing the genericity condition on the set of connections, defined in terms of the Riemann curvature tensor. This shall enable us to investigate the structure of the space of local parallel metrics; that is, metrics defined on an open set of the manifold, which are parallel with respect to the given Levi-Civita connection [nabla]. This, in turn, shall lead to a decomposition of the tangent space of the manifold into a direct sum of orthogonal subbundles. The original metric h, restricted to the subbundles of the decomposition, defines a set of positive semi-definite tensor fields [h.sub.a]. It is proved that a metric is parallel with respect to [nabla] if and only if it is a positive-definite linear combination of the [h.sub.a]. Lastly, we provide an example that illustrates the method described herein.

In [1], it is determined when an arbitrary analytic symmetric connection is a Levi-Civita connection for some metric. The problem of finding the metric from the Ricci curvature has been studied by DeTurk [2], [3], [4] and [5], and from the curvature in general relativity by Hall [6] and Hall and McIntosh [7].

2. Parallel metrics for generic Levi-Civita connections

In what follows, M shall denote a connected manifold of dimension n. This is not essential but it shall lead to concepts that are more familiar. Moreover, the scope of the paper is not thereby limited since the results below may be applied to the separate components of a general manifold.

It will also be convenient to represent a metric in terms of contravariant indices:

<,> = g = [n.summation over (i,j=1)] [g.sup.ij][X.sub.i] [cross product] [X.sub.j] where ([X.sub.1], ..., [X.sub.n]) is a local frame. This shall be the convention throughout.

Define a connection [nabla] on M to be generic at a point m [member of] M if there exist tangent vectors [[xi].sub.1], [[xi].sub.2] [member of] [T.sub.m]M such that the linear transformation

R([[xi].sub.1], [[xi].sub.2]): [T.sub.m]M [right arrow] [T.sub.m]M

has n distinct (complex) eigenvalues, where R denotes the Riemann curvature tensor of [nabla]. A connection is generic if it is generic at every point m [member of] M. Henceforth [nabla] shall denote a generic symmetric connection on M with parallel metric h.

Consider a connected open set U of M for which there exist vector fields [[xi].sub.1], [[xi].sub.2] on U such that

R([[xi].sub.1](u), [[xi].sub.2](u)): [T.sub.u]M [right arrow] [T.sub.u]M

has n distinct eigenvalues [[lambda].sub.i] = [[lambda].sub.i](u), for each u [member of] U, and an associated frame of (complex) eigenvector fields ([Z.sub.1], ...,[Z.sub.n]) defined on U:

R([[xi].sub.1], [[xi].sub.2])([Z.sub.i]) = [[lambda].sub.i][Z.sub.i]

Let [??] denote the set of all such connected open sets of M. Since [nabla] is generic, [??] is an open covering of M. If [nabla] [member of] [??] then any connected open subset W [subset or equal to] V is also in [??].

Lemma 1

(i) The frame of eigenvector fields ([Z.sub.1], ..., [Z.sub.n]) of R([[xi].sub.1], [[xi].sub.2]), after a possible reordering, has the form

[X.sub.1] + i[X.sub.2], [X.sub.1] - i[X.sub.2], ..., [X.sub.2m-1] + i[X.sub.2m], [X.sub.2m-1] - i[X.sub.2m], ([X.sub.n])

where the [X.sub.i] are vector fields on U and [X.sub.n], the eigenvector field corresponding to the zero eigenvalue, is included if n is odd. Furthermore, (ii) [X.sub.1], ..., [X.sub.2m], ([X.sub.n]) is an orthogonal frame on U for any metric g on U, parallel with respect to [nabla]. That is, g is expressible in the form g = [n.summation over (i=1)] [g.sub.i][X.sub.i] [cross product] [X.sub.i] for some functions [g.sub.i]: U [right arrow] [[Real part].sup.+].

Proof:

(i) Since [nabla] is the Levi-Civita connection of the metric h, there exists an orthonormal basis of [T.sub.u]M at each u [member of] U, with respect to which R([[xi].sub.1], [[xi].sub.2]) is represented as a skew-symmetric matrix. Hence the eigenvalues [[lambda].sub.i] = [[lambda].sub.i](u) are purely imaginary with associated eigenvector fields [Z.sub.2k-1] = [X.sub.2k-1] + i[X.sub.2k] for [[lambda].sub.2k-1] and [Z.sub.2k] = [[bar.Z].sub.2k-1] = [X.sub.2k-1] - i[X.sub.2k] for [[lambda].sub.2k] = -[[lambda].sub.2k-1], 1 [less than or equal to] k [less than or equal to] m, except for [[lambda].sub.n] = 0 when n is odd, with associated eigenvector field [Z.sub.n] = [X.sub.n].

(ii) Let g = [n.summation over (i,j=1)] [g.sup.ij][Z.sub.i] [cross product] [Z.sub.j] be a metric parallel on U; thus R([[xi].sub.1], [[xi].sub.2])(g) = 0. The explicit form of g gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore [g.sup.ij]([[lambda].sub.i] + [[lambda].sub.j]) = 0 for all 1 [less than or equal to] i, j [less than or equal to] n. The eigenvalues [[lambda].sub.i] are distinct, by hypothesis, and [[lambda].sub.2k] = -[[lambda].sub.2k-1], 1 [less than or equal to] k [less than or equal to] m, except for [[lambda].sub.n] = 0 when n is odd. Hence [g.sup.ij] = 0, unless (i, j) = (2k - 1, 2k) or (i, j) = (2k, 2k - 1) for some k [member of] {1, ..., m}, or i = j = n when n is odd. It follows that [g.sup.ij] is block diagonal with 2 x 2 blocks down the main diagonal and with a single 1 x 1 block for odd n. In the ([X.sub.1], ..., [X.sub.n]) frame the kth 2 x 2 block has the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [g.sub.2k-1] = [g.sub.2k]:= [g.sup.2k-1,2k] + [g.sup.2k,2k-1]. Defining [g.sub.n] := [g.sup.nn] for odd n,

g = [n.summation over (i=1)][g.sub.i][X.sub.i] [cross product] [X.sub.i]

q.e.d.

Let [theta] = [[theta].sup.i.sub.j] denote the connection form of [nabla] in the ([X.sub.1], ..., [X.sub.n]) frame:

[nabla] [X.sub.j] = [n.summation over (i=1)][X.sub.i] [cross product] [[theta].sup.i.sub.j]

The following lemma indicates the amount of variation allowed among parallel metrics.

Lemma 2 Let <, > [sup.1] = [n.summation over (i=1)][g.sub.i][X.sub.i] [cross product] [X.sub.i] and <,> [sup.2] = [n.summation over (i=1)] [k.sub.i][X.sub.i] [cross product] [X.sub.i] be two arbitrary metrics on U, parallel with respect to [nabla]. Then there exist constants [c.sub.i] [member of] [[Real part].sup.+] such that [g.sub.i] = [c.sub.i][k.sub.i], for all 1 [less than or equal to] i [less than or equal to] n.

Proof:

Since [<, >.sup.1] is parallel,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When i = j this gives, [[theta].sup.i.sub.i] = -1/2[dlogg.sub.i]. Similarly, [[theta].sup.i.sub.i] = - [1/2][dlogk.sub.i]. Therefore

[dlogg.sub.i] = [dlogk.sub.i]

and since U is connected, [g.sub.i] = [c.sub.i][k.sub.i] for some [c.sub.i] [member of] [[Real part].sup.+].

q.e.d.

Next, we shall seek to make this observation stronger. In light of Lemma 1, the metric h can be written locally as

h[|.sub.U] = [n.summation over (i=1)][[rho].sub.i][X.sub.i] [cross product] [X.sub.i]

for some functions [[rho].sub.i]: U [right arrow] [[Real part].sup.+]. Define the orthonormal basis of local vector fields ([Y.sub.1], ..., [Y.sub.n]) by [Y.sub.i] := [square root of [[rho].sub.i]][X.sub.i], for 1 [less than or equal to] i [less than or equal to] n. Then h[|.sub.U] = [Y.sub.1] [cross product] [Y.sub.1] + ... + [Y.sub.n] [cross product] [Y.sub.n]. Let [omega] = [[omega].sup.i.sub.j] denote the connection form of [nabla] with respect to the frame ([Y.sub.1], ..., [Y.sub.n]):

[nabla][Y.sub.j] = [n.summation over (i=1)][Y.sub.i] [cross product] [[omega].sup.i.sub.j]

Lemma 3 Consider g = [n.summation over (i=1)][c.sub.i][Y.sub.i] [cross product] [Y.sub.i], where [c.sub.i] [member of] [Real part]. [nabla]g = 0 if and only if ([c.sub.i] - [c.sub.j]) [[omega].sup.i.sub.j](u) = 0 for all u [member of] U and 1 [less than or equal to] i, j [less than or equal to] n.

Proof:

Taking the covariant derivative of g gives,

[nabla]g = [nabla] ([n.summation over (i=1)][c.sub.i][Y.sub.i] [cross product] [Y.sub.i]) = [n.summation over (i,j=1)] [Y.sub.i] [cross product] [Y.sub.j] [cross product] ([c.sub.j][[omega].sup.i.sub.j] + [c.sub.i][[omega].sup.j.sub.i])

Therefore [nabla]g = 0 if and only if [c.sub.j][[omega].sup.i.sub.j] + [c.sub.i][[omega].sup.j.sub.i] = 0 for all 1 [less than or equal to] i, j [less than or equal to] n. Since [[omega].sup.i.sub.j] = -[[omega].sup.j.sub.i], this holds if and only if ([c.sub.i] - [c.sub.j])[[omega].sup.i.sub.j] = 0 for all 1 [less than or equal to] i, j [less than or equal to] n.

q.e.d.

Define r(U) to be the equivalence relation on {1, ..., n} generated by the relations {(i, j)|[w.sup.i.sub.j](u) [not equal to] 0 for some u [member of] U}. Thus ir(U)j for i [not equal to] j if and only if there exists a sequence i = [i.sub.1], ..., [i.sub.k] = j in {1, ...,n} and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] partitions {1, ...,n} into [beta](U) disjoint subsets [P.sub.1](U), [P.sub.[beta](U)](U).

Define tensor fields [h.sub.i](U) on U by

[h.sub.i](U) := [summation over (j[member of][P.sub.i](U))] [Y.sub.j] [cross product] [Y.sub.j],

for 1 [less than or equal to] i [less than or equal to] [beta](U). Note that

h[|.sub.U] = [[beta](U).summation over (i=1)] [h.sub.i](U)

Lemma 4 If

g = [[beta](U).summation over (i=1)] [a.sub.i][h.sub.i](U)

for constants [a.sub.i] [member of] [Real part] then g is parallel with respect to [nabla].

Proof:

Suppose that g = [n.summation over (i=1)][c.sub.i][Y.sub.i] [cross product] [Y.sub.i] = [[beta](U).summation over (k=1)] [a.sub.k][h.sub.k](U) for constants [c.sub.i], [a.sub.k] [member of] [Real part]. If ir(U)j then i and j belong to the same equivalence class [P.sub.k](U), say. Hence [c.sub.i] = [c.sub.j] = [a.sub.k]. On the other hand, if i and j are not r(U)-related then [[omega].sup.i.sub.j](u) = 0 for all u [member of] U. In either case, ([c.sub.i] - [c.sub.j]) [[omega].sup.i.sub.j](u) = 0 for all u [member of] U. Therefore by Lemma 3, g is parallel with respect to [nabla].

q.e.d.

Lemma 5 If g is a metric on U parallel with respect to [nabla] then

g = [[beta](U).summation over (i=1)] [a.sub.i][h.sub.i](U)

for some constants [a.sub.i] [member of] [[Real part].sup.+].

Proof:

Let g be a metric on U parallel with respect to [nabla]. By Lemma 1, g may be written g = [n.summation over (i=1)][g.sub.i][X.sub.i] [cross product] [X.sub.i] for some functions [g.sub.i]: U [right arrow] [[Real part].sup.+]. Then Lemma 2 and the fact that h[|.sub.U] = [n.summation over (i=1)] [[rho].sub.i][X.sub.i] [cross product] [X.sub.i] imply that g is of the form

g = [n.summation over (i=1)][c.sub.i][[rho].sub.i][X.sub.i] [cross product] [X.sub.i] = [n.summation over (i=1)][c.sub.i][Y.sub.i] [cross product] [Y.sub.i]

for some constants [c.sub.i] [member of] [[Real part].sup.+]. By Lemma 3, ([c.sub.i] - [c.sub.j]) [[omega].sup.i.sub.j](u) = 0 for all u [member of] U and 1 [less than or equal to] i, j [less than or equal to] n. Hence [c.sub.i] = [c.sub.j] whenever ir(U)j. This means that g may be expressed as g = [[beta](U).summation over (i=1)][a.sub.i][h.sub.i](U) for constants [a.sub.i] [member of] [[Real part].sup.+].

q.e.d.

Up to this point we have explored the structure of the set of parallel metrics on a single open set. We shall now investigate how these structures relate to each other on intersecting sets. The definition of [h.sub.i](U) described above depends upon (1) the choice of vector fields [[xi].sub.1] and [[xi].sub.2] on U enforcing the genericity condition, (2) the ordering of the associated orthonormal frame ([Y.sub.1], ...,[Y.sub.n]), and (3) the ordering of the blocks, [P.sub.1](U), [P.sub.[beta](U)](U), of the associated partition. For each U in [??] we shall require that such choices have been made and define the tensor fields [h.sub.i](U), 1 [less than or equal to] i [less than or equal to] [beta](U), accordingly.

Lemma 6 Let W [subset or equal to] U be sets in [??]. Then there exist constants [c.sub.iq] [member of] [Real part] such that

[h.sub.i](U)[|.sub.W] = [[beta](W).summation over (q=1)] [c.sub.iq][h.sub.q](W)

for 1 [less than or equal to] i [less than or equal to] [beta](U).

Proof:

By Lemma 5,

[[beta](U).summation over (j=1)] [h.sub.j](U)[|.sub.W] = [[beta](W).summation over (q=1)] [a.sub.q][h.sub.q](W)

for some [a.sub.q] [member of] [[Real part].sup.+]. By Lemmas 4 and 5,

[h.sub.i](U)[|.sub.W] + [[beta](U).summation over (j=1)][h.sub.j](U)[|.sub.W] = [[beta](U).summation over (q=1)] [b.sub.iq][h.sub.q](W)

for some [b.sub.iq] [member of] [[Real part].sup.+]. Subtracting gives

[h.sub.i](U)[|.sub.W] = [[beta](W).summation over (q=1)][c.sub.iq][h.sub.q](W)

where [c.sub.iq] := [b.sub.iq] - [a.sub.q] [member of] [Real part].

q.e.d.

Let [[bar.Q].sub.i](U) be the subbundle of TU spanned by the vector fields {[Y.sub.j]: j [member of] [P.sub.i](U)}, for each i [member of] 1, ..., [beta](U). We then have a decomposition of the tangent space TU = [[bar.Q].sub.1](U) [cross product]...[cross product][[bar.Q].sub.[beta](U)](U). Denote the dual subbundles by [[bar.Q].sup.*.sub.i](U). Then [T.sup.*]U = [[bar.Q].sup.*.sub.1](U) [cross product] ... [cross product] [[bar.Q].sup.*.sub.[beta](U)](U). Observe that [h.sub.i](U) is a section of [[bar.Q].sub.i](U) [cross product] [[bar.Q].sub.i](U), defining a positive semi-definite bilinear form [h.sub.i](U): [T.sup.*.sub.u]U x [T.sup.*.sub.u]U [right arrow] [Real part] for each u [member of] U. Set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Lemma 7 Let W [subset or equal to] U be sets in [??]. Then there exists a partition

[GAMMA](1), ...,[GAMMA]([beta](U))

of the set of integers {1, ..., [beta](W)} such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[h.sub.i](U)[|.sub.W] = [summation over (q[member of][GAMMA](i))][h.sub.q](W)

for 1 [less than or equal to] i [less than or equal to] [beta](U).

Proof:

By Lemma 6, there exist constants [c.sub.iq] [member of] [Real part] such that

[h.sub.i](U)[|.sub.W] = [[beta](W).summation over (q=1)] [c.sub.iq][h.sub.q](W)

for 1 [less than or equal to] i [less than or equal to] [beta](U). Define subsets [gamma](i) and [GAMMA](i) of {1, ..., [beta](W)} by

[gamma](i) := {q: [c.sub.iq] [not equal to] 0}

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for 1 [less than or equal to] i [less than or equal to] [beta](U). Now,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows that [GAMMA](1), ..., [GAMMA]([beta](U)) is a partition of {1, ...,[beta](W)} and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for 1 [less than or equal to] i [less than or equal to] [beta](U).

Furthermore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

since [c.sub.iq] = 0 for q [member of] [GAMMA](j) when j [not equal to] i. Consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore [c.sub.iq] = 1 for all q [member of] [GAMMA](i) and so

[h.sub.i](U)[|.sub.W] = [summation over (q[member of][GAMMA](i))] [h.sub.q](W)

q.e.d.

We now turn to the problem of how to piece together these local parallel metrics into a global one. This is done by means of an appropriately defined equivalence relation. Let U be any subset of [??] that covers M and has the property that the intersection of pairs of sets in U is connected. For instance, we can choose U to be a good refinement of [??]. Let I := {(i, U): U [member of] U and 1 [less than or equal to] i [less than or equal to] [beta](U)} and define ~ to be the equivalence relation on I generated by the relations

(i,U)~(j,V) if U [intersection] V [not equal to] 0 and [[bar.Q].sub.i](U) [intersection] [[bar.Q].sub.j](V) [not equal to] 0

where "0" means the zero distribution on U [intersection] V. These relations are required in order to join the [[bar.Q].sub.i] distributions together in a smooth way. The equivalence partitions I into A blocks denoted [I.sub.1], ...,[I.sub.A].

Lemma 8 For each a [member of] {1, ..., A} and U [member of] U the set {i: (i, U) [member of] [I.sub.a]} is non-empty.

Proof:

Let C(U) denote the subset of U consisting of all sets U' for which there exists a sequence of sets U' = [U.sub.1], [U.sub.2], ...,[U.sub.k] = U in U such that [U.sub.l] [intersection] [U.sub.l+1] [not equal to] 0 for all 1 [less than or equal to] l [less than or equal to] k-1. We claim that C(U) = U. First observe that if [nabla] [member of] U and V [intersection] U' [not equal to] 0 for some U' [member of] C(U) then V [member of] C(U) also. Let S denote the union of the sets in C(U). Suppose that S is not equal to M. Since M is connected the boundary [partial derivative]S, of S is non-empty. U covers M, so there exists an open set V [member of] U such that V [intersection] [partial derivative]S [not equal to] 0. This means that V [intersection] S [not equal to] 0 and V [intersection] [S.sup.c] [not equal to] 0, where [S.sup.c] denotes the compliment of S in M. Hence V [intersection] U' [not equal to] 0 for some U' [member of] C(U) and V [not member of] C(U), which is a contradiction. Therefore S = M. Let W [member of] U. Then W [intersection] U' [not equal to] 0 for some U' [member of] C(U) and so W [member of] C(U). This demonstrates the claim.

Let ([i.sub.1], [U.sub.1]) be any representative of [I.sub.a]. We have shown that there exists a sequence of sets [U.sub.2], ..., [U.sub.k] = U in U such that [U.sub.l] [intersection] [U.sub.l+1] [not equal to] 0 for all 1 [less than or equal to] l [less than or equal to] k-1. For each l [member of] {1, ...,k - 1} and p [member of] {1, ..., [beta]([U.sub.l])} there exists at least one q [member of] {1, ..., [beta]([U.sub.l+1])} such that [[bar.Q].sub.P]([U.sub.l]) [intersection] [[bar.Q].sub.q]([U.sub.l+1]) [not equal to] 0. Therefore we can find a sequence [i.sub.2], ..., [i.sub.k] such that ([i.sub.1], [U.sub.1]) ~ ([i.sub.2], [U.sub.2]) ~ ... ~ ([i.sub.k], [U.sub.k]) = ([i.sub.k], U). Hence [i.sub.k] [member of] {i: (i, U) [member of] [I.sub.a]}.

q.e.d.

By Lemma 8, we may define a non-trivial distribution [Q.sub.a](U) on U by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all a [member of] {1, ...,A} and U [member of] U. Define the distributions [Q.sub.a] on M by specifying their restriction on each U [member of] U to be

[Q.sub.a][|.sub.U] := [Q.sub.a](U)

Lemma 9 The [Q.sub.a] are well-defined.

Proof:

Let V be another set in U having non-zero intersection with U. We must show that on the intersection, [Q.sub.a](U)[|.sub.U[intersection]V] = [Q.sub.a](V)[|.sub.U[intersection]V]. By Lemma 7, the set {1, ..., [beta](U [intersection] V)} partitions into [[GAMMA].sub.U](1), ..., [[GAMMA].sub.U]([beta](U)) in such away that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

{1, ..., [beta](U [intersection] V)} also partitions into [[GAMMA].sub.V](1), ..., [[GAMMA].sub.V]([beta](V)) in such away that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Observe that if [[GAMMA].sub.U](i) [intersection] [[GAMMA].sub.V](j) [not equal to] 0 then [[bar.Q].sub.i](U)[|.sub.U[intersection]V] [intersection] [[bar.Q].sub.j](V)[|.sub.U[intersection]V] [not equal to] 0 and so (i, U) ~ (j, V). Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly, [Q.sub.a](V)[|.sub.U[intersection]V] [subset or equal to] [Q.sub.a](U)[|.sub.U[intersection]V] and so the two distributions are equal on U [intersection] V.

q.e.d.

The tangent space has the direct sum decomposition into subbundles:

TM = [Q.sub.1] [direct sum] ... [direct sum] [Q.sub.A]

Denote the dual subbundles by [Q.sup.*.sub.a] and the fibre of [Q.sup.*.sub.a] over m [member of] M by [Q.sup.*.sub.a](m). Then the cotangent space at m has the decomposition:

[T.sup.*.sub.m]M = [Q.sup.*.sub.1](m) [direct sum] ... [direct sum] [Q.sup.*.sub.A](m)

h is a section of TM [cross product] TM and therefore determines a bilinear map h(m): [T.sup.*.sub.m]M x [T.sup.*.sub.m]M [right arrow] [Real part] for each m [member of] M. Define positive semi-definite metrics [h.sub.a] on M for 1 [less than or equal to] a [less than or equal to] A by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for each m [member of] M. On U [member of] U,

[h.sub.a][|.sub.U] = [summation over ({k:(k,U)[member of][I.sub.a]})] [h.sub.k](U)

Therefore h = [A.summation over (a=1)] [h.sub.a].

We may now describe the parallel metrics of [nabla] on M.

Theorem 10 g is a parallel metric on M if and only if it can be written as g = [A.summation over (a=1)][c.sub.a][h.sub.a] for some [c.sub.a] [member of] [[Real part].sup.+].

Proof:

[??] By Lemma 4 and the fact that [h.sub.a][|.sub.U] = [summation over ({k:(k,U)[member of][I.sub.a]})] [h.sub.k](U), [nabla][h.sub.a] = 0 for all 1 [less than or equal to] a [less than or equal to] A.

Therefore any g = [A.summation over (a=1)] [c.sub.a][h.sub.a], where [c.sub.a] [member of] [[Real part].sup.+], is a parallel metric on M.

[??] Suppose that g is a parallel metric on M. Let U and V be two sets in U with non-empty intersection. By Lemma 7, the set {1, ..., [beta](U [intersection] V)} partitions into [[GAMMA].sub.U](1), ..., [[GAMMA].sub.U]([beta](U)) in such a way that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[h.sub.i](U)[|.sub.U[intersection]V] = [summation over (q[member of][[GAMMA].sub.U](i))] [h.sub.q](U [intersection] V)

{1, ..., [beta](U [intersection] V)} also partitions into [[GAMMA].sub.V](1), ..., [[GAMMA].sub.V]([beta](V)) in such a way that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[h.sub.j](V)[|.sub.U[intersection]V] = [summation over (q[member of][[GAMMA].sub.V](j))] [h.sub.q] (U [intersection] V)

Suppose that [[bar.Q].sub.i](U) [intersection] [[bar.Q].sub.j](V) [not equal to] 0 for some i and j. Then there exists an element p [member of] [[GAMMA].sub.U](i) [intersection] [[GAMMA].sub.V](j).

For any W [member of] U, Lemma 5 defines positive constants [c.sub.k](W) by

g[|.sub.W] = [[beta](W).summation over (k=1)] [c.sub.k](W)[h.sub.k](W)

Therefore on U [intersection] V, g may be expressed as

g[|.sub.U[intersection]V] = [[beta](U).summation over (k=1)] [c.sub.k](U)[h.sub.k](U)[|.sub.U[intersection]V] = [[beta](U).summation over (k=1)] [c.sub.k](U) [summation over (q[member of][[GAMMA].sub.U](k))] [h.sub.q](U [intersection] V)

Similarly,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The coefficient of [h.sub.p](U [intersection] V) is [c.sub.i](U) = [c.sub.j](V). Therefore

[[bar.Q].sub.i](U) [intersection] [[bar.Q].sub.j](V) [not equal to] 0 [??] [c.sub.i](U) = [c.sub.j](V)

From the definition of the equivalence ~ it follows that

(i, [U.sub.1]) ~ (j, [U.sub.2]) [??] [c.sub.i]([U.sub.1]) = [c.sub.j]([U.sub.2])

This allows us to define the positive constants [c.sub.a] := [c.sub.k](U), where (k, U) is any representative of [I.sub.a], for 1 [less than or equal to] a [less than or equal to] A.

Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows that g = [A.summation over (a=1)] [c.sub.a][h.sub.a].

q.e.d.

Corollary 11 The manifold of parallel metrics on M has dimension A.

Given the Riemannian manifold (M, h) with generic Levi-Civita connection [nabla], the determination of the [h.sub.a] is an algebraic construction. Therefore Theorem 10 enables one to obtain all parallel metrics of [nabla] by purely algebraic means; integration of differential equations is unnecessary.

Let us summarize the steps in the procedure.

1. Find an open cover U [subset or equal to] [??] of M having the property that the intersection of pairs of sets in U is connected. For each U [member of] U, follow steps 2-5:

2. Find a frame of eigenvector fields ([Z.sub.1], ..., [Z.sub.n]) on U for R([[xi].sub.1], [[xi].sub.2]).

3. Obtain the associated frame of orthogonal vector fields ([X.sub.1], ..., [X.sub.n]).

4. Define the orthonormal frame ([Y.sub.1], ..., [Y.sub.n]).

5. Obtain the decomposition TU = [[bar.Q].sub.1](U) [direct sum] ... [direct sum] [[bar.Q].sub.[beta](U)](U).

6. Construct the subbundles [Q.sub.a].

7. Define the tensor fields [h.sub.a] and apply the theorem.

Example Consider the Riemannian manifold (M, h) where M := [[Real part].sup.4] and h := [dx.sup.2] + [e.sup.2x][dy.sup.2] + [du.sup.2] + [e.sup.2u][dv.sup.2], which in contravariant form is [[partial derivative]/[partial derivative]x] [cross product] [[partial derivative]/[partial derivative]x] + [e.sup.-2x] [[partial derivative]/[partial derivative]y] [cross product] [[partial derivative]/[partial derivative]y] + [[partial derivative]/[partial derivative]u] [cross product] [[partial derivative]/[partial derivative]u] + [e.sup.-2u] [[partial derivative]/[partial derivative]v] [cross product] [[partial derivative]/[partial derivative]v]. (M, h) is a Cartesian product of two isomorphic irreducible Riemannian manifolds and therefore it is expected that the general parallel metric would be a positive-definite linear combination of the pullbacks onto M of the component metrics, [[partial derivative]/[partial derivative]x] [cross product] [[partial derivative]/[partial derivative]x] + [e.sup.-2x] [[partial derivative]/[partial derivative]y] [cross product] [[partial derivative]/[partial derivative]y] and [[partial derivative]/[partial derivative]u] [cross product] [[partial derivative]/[partial derivative]u] + [e.sup.-2u] [[partial derivative]/[partial derivative]v] [cross product] [[partial derivative]/[partial derivative]v].

The Christoffel symbols for the Levi-Civita connection [nabla] of h are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and all others zero. Define a good cover U := {U, U'} of M by

U := {(x,y,u,v) [member of] [[Real part].sup.4]: x > u}

U' := {(x,y,u,v) [member of] [[Real part].sup.4]: x < u + log2}

On U let [xi]1 := [[partial derivative]/[partial derivative]x] + [[partial derivative]/[partial derivative]u] and [xi]2 := [[partial derivative]/[partial derivative]y] + [[partial derivative]/[partial derivative]v]. The Riemann curvature with respect to the frame ([[partial derivative]/[partial derivative]x], [[partial derivative]/[partial derivative]y], [[partial derivative]/[partial derivative]u], [[partial derivative]/[partial derivative]v]) on U is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The eigenvalues are [lambda] = -[ie.sup.x], [ie.sup.x], -[ie.sup.u] and [ie.sup.u], which are distinct on U. Corresponding eigenvector fields on U are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which defines

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The orthonormal frame ([Y.sub.1], ..., [Y.sub.4]) is then given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The curvature form [omega] = [[omega].sup.i.sub.j] with respect to ([Y.sub.1], ...,[Y.sub.4]) satisfies ([[omega].sup.2.sub.1]([Y.sub.2]) = [[omega].sup.4.sub.3]([Y.sub.4]) = 1. Therefore 2r(U)1 and 4r(U)3. Furthermore, = [[omega].sup.i.sub.j] = 0 for i [member of] {1, 2} and j [member of] {3, 4}. Thus there are exactly two equivalence classes for the equivalence relation r(U):

[P.sub.1](U) = {1, 2} and [P.sub.2](U) = {3, 4}

This gives,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On U' let [[xi]'.sub.1] := [[partial derivative]/[partial derivative]x] + 2[[partial derivative]/[partial derivative]u] and [[xi]'.sub.2] := [[partial derivative]/[partial derivative]y] + [[partial derivative]/[partial derivative]v]. The Riemann curvature with respect to the frame ([[partial derivative]/[partial derivative]x], [[partial derivative]/[partial derivative]y], [[partial derivative]/[partial derivative]u], [[partial derivative]/[partial derivative]v]) on U' is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The eigenvalues are [lambda]' = -[ie.sup.x], [ie.sup.x], -2[ie.sup.u] and 2[ie.sup.u], which are distinct on U'. Corresponding eigenvector fields on U' are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which defines

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The orthonormal frame ([Y'.sub.1], ..., [Y'.sub.4]) is then given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Continuing the analysis as above gives

[P.sub.1](U') = {1, 2} and [P.sub.2](U') = {3, 4}

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Next we consider the equivalence relation ~. Restricted to U [intersection] U',

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore (1, U) ~ (1, U'), (2,U) ~ (2,U') and there are no other non-trivial relations.

This gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By Theorem 10, g is a parallel metric with respect to [nabla] if and only if g = [c.sub.1][h.sub.1] + [c.sub.2][h.sub.2] for some constants [c.sub.1], [c.sub.2] [member of] [[Real part].sup.+]; the anticipated result.

References

[1] R. Atkins, When is a Connection a Metric Connection? New Zealand Journal of Mathematics, Volume 38 (2008), 225-238.

[2] D. DeTurck, Metrics with prescribed Ricci curvature, in Seminar on Differential Geometry, ed S.T. Yau, Annals of Math Studies 102, Princton university Press, Princton, 1982, 525-537.

[3] D. DeTurck, Existence of Metrics with prescribed Ricci curvature. Local Theory, Inventiones Math. 65, 1981, 179-201.

[4] D. DeTurck, he Cauchy problem for Lorentz metrics with prescibed Ricci curvature, Compositio Math 48, 1983, 327-349.

[5] D. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Differential Geometry 18, 1983, 157-162.

[6] G.S. Hall, Curvature colliniations and the determination of the metric from the curvature in general relativity, Gen. Relativity Gravitation, 15, 1983, 581.

[7] G.S. Hall and C.B.G. McIntosh, Algebraic determination of the metric from the curvature in general relativity, Internat. J. Theoret. Phys. 22, 1983, 469

Richard Atkins

Department of Mathematics

Trinity Western University

7600 Glover Road Langley, BC, V2Y1Y1 Canada

richard.atkins@twu.ca

tel. 1-604-888-7511
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