New York Journal of Mathematics

New York J. Math. 15 (2009) 283–289.

Conditions for the algebraic determination of the metric from the

curvature

Richard Atkins

Abstract. We establish new conditions ensuring that a Riemannian metric may be constructed, up to a conformal factor, from the skewsymmetries of its Riemann curvature tensor.

Contents

1. Introduction 283 2. Eigenvectors of the curvature 284 References 289

1. Introduction

The problem of constructing the metric g = g_{ij}, up to a conformal factor, from the Riemann curvature R = R_{jkl}^{i} has been investigated by Ihrig ([5], [6]). The method hinges on the fact that the lowered curvature tensor R_{ijkl} is skew-symmetric in the indices i and j. Specifically, on an n-dimensional manifold M, the components g_{ij} of the metric may be obtained from the set of linear equations

∑_{s=1}^{n}(g_{is}R_{jkl}^{s} + g_{js}R_{ikl}^{s}) = 0 (1)

for 1 ≤ i, j, k, l ≤ n, provided the solutions g = g_{ij}, consisting of symmetric two-tensors, span a vector space of dimension one. The one-dimensionality of solutions is guaranteed by a condition Ihrig terms total. The Riemann tensor is total at a point m ∈ M if

{R(u, v) : u, v ∈ T_{m}M}

283^{ISSN 1076-9803/09}

Received April 10, 2008.

Mathematics Subject Classification. 53B21.

Key words and phrases. Riemann metric, curvature tensor, total.

generates a vector space of dimension n(n − 1)/2. This is equivalent to requiring that the dimension of the space generated by the two-forms R(∗, ∗)_{j}^{i}, defined by the Riemann curvature, is equal to n(n − 1)/2. This is a rather strong condition to impose; for many examples of interest, equations (1) are sufficient to determine the metric even when the totality requirement is not satisfied. McIntosh and Halford ([8]) have shown that this condition can be weakened for the case of a metric of type (1,3), in that it is sufficient to demand that the dimension of the space generated by the two-forms R(∗, ∗)_{j}^{i} be greater than three. A shorter, geometrical proof of this result was obtained by Hall and McIntosh ([4]).

In this paper we provide a sufficient condition of a somewhat different nature for the determination of the metric by equations (1) (up to conformal equivalence). These will appear to be relatively mild in comparison to assumptions based upon the dimension of the space of Riemann two-forms. On the other hand, the latter applies to pseudo-Riemannian metrics of all signatures whereas our approach is valid only for positive definite metrics. In addition, our condition is somewhat more complicated to state.

We shall consider two prerequisites for the Riemann tensor. First, R(ξ_{1}, ξ_{2}) must possess distinct eigenvalues for some choice of tangent vectors ξ_{1}, ξ_{2} ∈ T_{m}M and second, R(∗, ∗) must be sufficiently nonzero at m, in a sense to be made precise below. It will then follow that the metric can be recovered from the symmetric solutions to (1), up to a conformal scale.

2. Eigenvectors of the curvature

Suppose we are given the Riemann curvature R(∗, ∗) of an unknown positive definite metric g on a manifold M of dimension n. For tangent vectors ξ_{1}, ξ_{2} ∈ T_{m}M, R(ξ_{1}, ξ_{2}) is nondegenerate at m if the associated linear transformation

R(ξ_{1}, ξ_{2}) : T_{m}M → T_{m}M

has n distinct (complex) eigenvalues. We shall assume this to be so and denote by (Z_{1}, …, Z_{n}) an eigenbasis of R(ξ_{1}, ξ_{2}), with corresponding distinct eigenvalues (λ_{1}, …, λ_{n}), respectively:

R(ξ_{1}, ξ_{2})(Z_{i}) = λ_{i}Z_{i}

for 1 ≤ i ≤ n.

The following lemma is the salient observation.

Lemma 1. (i) The eigenbasis (Z_{1}, …, Z_{n}) of R(ξ_{1}, ξ_{2}), after a possible reordering, has the form

X_{1} + iX_{2}, X_{1} − iX_{2}, …, X_{2k−1} + iX_{2k}, X_{2k−1} − iX_{2k}, (X_{n})

where the X_{i} are vectors in T_{m}M and X_{n}, the eigenvector corresponding to the zero eigenvalue, is included if n is odd.

(ii) (X_{1}, …, X_{2k}, (X_{n})) is an orthogonal basis for T_{m}M. That is, g(m) is expressible in the form g(m) = ∑_{i=1}^{n} g_{i}θ^{i} ⊗ θ^{i} for some constants g_{i} ∈ ℜ^{+}, where

(θ^{1}, …, θ^{2k}, (θ^{n}))

is the basis of covectors dual to (X_{1}, …, X_{2k}, (X_{n})).

Proof. (i) Since ∇ is the Levi-Civita connection of the Riemannian metric g, there exists an orthonormal basis of T_{m}M, at each m ∈ M, with respect to which R(ξ_{1}, ξ_{2}) is represented as a skew-symmetric matrix. Hence the eigenvalues λ_{i} = λ_{i}(m) are purely imaginary with associated eigenvectors Z_{2l−1} = X_{2l−1} + iX_{2l} for λ_{2l−1} and Z_{2l} = Z_{2l−1} = X_{2l−1} − iX_{2l} for λ_{2l} = −λ_{2l−1}, 1 ≤ l ≤ k, except for λ_{n} = 0 when n is odd, with eigenvector Z_{n} = X_{n}.

(ii) Let g = ∑_{i,j=1}^{n} g^{ij}Z_{i} ⊗ Z_{j} be a parallel metric on M, written contravariantly; thus R(ξ_{1}, ξ_{2})(g) = 0. The explicit form of g gives

R(ξ_{1}, ξ_{2})(g) = [∇_{ξ_{1}}, ∇_{ξ_{2}}](g) − ∇_{[ξ_{1},ξ_{2}]}(g)

= ∑_{i,j=1}^{n} g^{ij}(R(ξ_{1}, ξ_{2})(Z_{i}) ⊗ Z_{j} + Z_{i} ⊗ R(ξ_{1}, ξ_{2})(Z_{j}))

= ∑_{i,j=1}^{n} g^{ij}(λ_{i}Z_{i} ⊗ Z_{j} + Z_{i} ⊗ λ_{j}Z_{j})

= ∑_{i,j=1}^{n} g^{ij}(λ_{i} + λ_{j})Z_{i} ⊗ Z_{j}.

Therefore g^{ij}(λ_{i}+λ_{j}) = 0 for all 1 ≤ i, j ≤ n. The eigenvalues λ_{i} are distinct, by hypothesis, and λ_{2l} = −λ_{2l−1}, 1 ≤ l ≤ k, except for λ_{n} = 0 when n is odd. Hence g^{ij} = 0, unless (i, j) = (2l − 1, 2l) or (i, j) = (2l, 2l − 1) for some l ∈ {1, …, k}, or i = j = n when n is odd. It follows that g^{ij} is block diagonal with 2 × 2 blocks down the main diagonal and with a single 1 × 1 block for odd n. In the (X_{1}, …, X_{n})-basis the l^{th} 2 × 2 block has the form

∑_{i,j=2l−1}^{2l} g^{ij}Z_{i} ⊗ Z_{j} = g^{2l−1,2l}Z_{2l−1} ⊗ Z_{2l} + g^{2l,2l−1}Z_{2l} ⊗ Z_{2l−1}
= g^{2l−1,2l}(X_{2l−1} + iX_{2l}) ⊗ (X_{2l−1} − iX_{2l})
+ g^{2l,2l−1}(X_{2l−1} − iX_{2l}) ⊗ (X_{2l−1} + iX_{2l})
= g_{2l−1}X_{2l−1} ⊗ X_{2l−1} + g_{2l}X_{2l} ⊗ X_{2l}

where g_{2l−1} = g_{2l} := g^{2l−1,2l} + g^{2l,2l−1}. Defining g_{n} := g^{nn} for odd n,

g = ∑_{i=1}^{n} g_{i}X_{i} ⊗ X_{i}.  □

The metric g at m is determined, by the lemma, up to the constants g_{i}. Denote by

h = ∑_{i=1}^{n} h_{i}θ^{i} ⊗ θ^{i}

the tentative form of g(m), where the h_{i} are positive constants to be ascertained.

Form the lowered Riemann curvature by contracting R(∗, ∗) = R(∗, ∗)_{j}^{i} with h:

R(∗, ∗)_{ij} := h_{i}R(∗, ∗)_{j}^{i}

where we have expressed the tensor indices in terms of the basis

(X_{1}, …, X_{2k}, (X_{n})).

The skew-symmetry of the lowered Riemann curvature gives

h_{i}R(∗, ∗)_{j}^{i} = −h_{j}R(∗, ∗)_{i}^{j} (2)

for all i, j, from which it follows that R(∗, ∗)_{j}^{i} is a nonzero two-form if and only if R(∗, ∗)_{i}^{j} is nonzero. Also, R(∗, ∗)_{i}^{i} = 0 for all i.

Let ∼ denote the equivalence relation on the set {1, …, n} generated by

i ∼ j if and only if R(∗, ∗)_{j}^{i} ≠ 0

for all i ≠ j. Thus i ∼ j if and only if i = j or there exists a sequence i = i_{1}, …, i_{r} = j such that i_{1} ≠ i_{2}, i_{2} ≠ i_{3}, i_{3} ≠ i_{4},…, i_{r−1} ≠ i_{r} and

R(∗, ∗)_{i_{l+1}}^{i_{l}} ≠ 0

for l = 1, …, r − 1. If i and j are related by such a sequence, h_{i} may be expressed in terms of h_{j} by means of the equations

h_{i_{1}}R(∗, ∗)_{i_{2}}^{i_{1}} = −h_{i_{2}}R(∗, ∗)_{i_{1}}^{i_{2}}
h_{i_{2}}R(∗, ∗)_{i_{3}}^{i_{2}} = −h_{i_{3}}R(∗, ∗)_{i_{2}}^{i_{3}}
h_{i_{3}}R(∗, ∗)_{i_{4}}^{i_{3}} = −h_{i_{4}}R(∗, ∗)_{i_{3}}^{i_{4}}
⋮
h_{i_{r−1}}R(∗, ∗)_{i_{r}}^{i_{r−1}} = −h_{i_{r}}R(∗, ∗)_{i_{r−1}}^{i_{r}}.

We shall say that the Riemann curvature possesses property P at m ∈ M if there exist ξ_{1}, ξ_{2} ∈ T_{m}M such that R(ξ_{1}, ξ_{2}) is nondegenerate and there is only one equivalence class: the entire set {1, …, n}. In this case, all the constants h_{1}, …, h_{n} may be determined from just one constant, say h_{1}. This gives g up to a conformal factor.

Theorem 2. A positive definite metric may be constructed algebraically, up to conformal equivalence, from its Riemann curvature by means of the equations

∑_{s=1}^{n}(g_{is}R_{jkl}^{s} + g_{js}R_{ikl}^{s}) = 0

for 1 ≤ i, j, k, l ≤ n, if the Riemann curvature satisfies property P.

Next, we provide examples that illustrate properties P and totality. In particular, we show that neither property generalizes the other.

Example 1. The 4-sphere: total but doesn’t satisfy P.

Consider any pair of linearly independent vectors (ξ_{1}, ξ_{2}) belonging to the same tangent space at some point of the unit 4-sphere. Apply the Gramm– Schmidt process to ξ_{1} and ξ_{2} to obtain vectors Y_{1} and Y_{2}. Now extend Y_{1} and Y_{2} to an orthonormal basis (Y_{1}, Y_{2}, Y_{3}, Y_{4}) of the tangent space, and denote by (ω^{1}, ω^{2}, ω^{3}, ω^{4}) the dual basis. With respect to this basis, the curvature form of the unit 4-sphere is

R(∗, ∗) = ( \begin{matrix}0 & ω^{1} ∧ ω^{2} & ω^{1} ∧ ω^{3} & ω^{1} ∧ ω^{4} \\ ω^{2} ∧ ω^{1} & 0 & ω^{2} ∧ ω^{3} & ω^{2} ∧ ω^{4} \\ ω^{3} ∧ ω^{1} & ω^{3} ∧ ω^{2} & 0 & ω^{3} ∧ ω^{4} \\ ω^{4} ∧ ω^{1} & ω^{4} ∧ ω^{2} & ω^{4} ∧ ω^{3} & 0 \end{matrix} )

from which it may be seen that the curvature is total. Furthermore, R(ξ_{1}, ξ_{2}) has rank two. Therefore λ = 0 is a degenerate eigenvalue of R(ξ_{1}, ξ_{2}) and so property P fails to hold.

Example 2. A conformally flat metric in 3-space: not total but satisfies P. Consider the metric g = e^{2xy}(dx^{2} + dy^{2} + dz^{2}) on

M := ℜ^{3} − {(x, y, z) : 2xy = 1}.

In the (∂_{x}, ∂_{y}, ∂_{z})-frame the curvature form is given by

R(∗, ∗) = ( \begin{matrix}0 & 0 & θ \\ 0 & 0 & ψ \\ −θ & −ψ & 0 \end{matrix} )

where

θ = −x^{2}dx ∧ dz + (−1 + xy)dy ∧ dz
ψ = (−1 + xy)dx ∧ dz − y^{2}dy ∧ dz

whence it is apparent that the totality condition fails to hold. We shall see, nevertheless, that property P satisfied. Choose ξ_{1} := ∂_{x} and ξ_{2} := ∂_{z}. Then

R(ξ_{1}, ξ_{2}) = ( \begin{matrix}0 & 0 & a \\ 0 & 0 & b \\ −a & −b & 0 \end{matrix} )

which has distinct eigenvalues

λ_{1} = ci, λ_{2} = −ci and λ_{3} = 0

where a := −x^{2}, b := −1 + xy and c := \sqrt{a^{2} + b^{2}}. Corresponding eigenvectors are

Z_{1} := ( \begin{matrix}−ai/c \\ −bi/c \\ 1 \end{matrix} ) Z_{2} := ( \begin{matrix}ai/c \\ bi/c \\ 1 \end{matrix} ) and Z_{3} := ( \begin{matrix}−b \\ a \\ 0 \end{matrix} ) .

This gives the orthogonal frame

X_{1} := ( \begin{matrix}0 \\ 0 \\ 1 \end{matrix} ) X_{2} := ( \begin{matrix}−a/c \\ −b/c \\ 0 \end{matrix} ) and X_{3} := ( \begin{matrix}−b \\ a \\ 0 \end{matrix} ) .

The curvature form in the (X_{1}, X_{2}, X_{3})-frame is

R(∗, ∗) = ( \begin{matrix}0 & α & β \\ −α & 0 & 0 \\ −β/c^{2} & 0 & 0 \end{matrix} )

where

α = (aθ + bψ)/c
β = bθ − aψ.

The two-form α is always nonzero and β is not a scalar multiple of α when 2xy ≠ 1. It follows that P holds on the manifold M.

Example 3. The 3-sphere: total and satisfies P.

Choose a local orthonormal frame (Y_{1}, Y_{2}, Y_{3}) of the 3-sphere and let (ω^{1}, ω^{2}, ω^{3}) be the dual coframe. The curvature form with respect to this frame is

R(∗, ∗) = ( \begin{matrix}0 & ω^{1} ∧ ω^{2} & ω^{1} ∧ ω^{3} \\ ω^{2} ∧ ω^{1} & 0 & ω^{2} ∧ ω^{3} \\ ω^{3} ∧ ω^{1} & ω^{3} ∧ ω^{2} & 0 \end{matrix} ) .

It is evident that the curvature is total. By choosing ξ_{1} = X_{1} and ξ_{2} = X_{2} at a point on the manifold we obtain

R(ξ_{1}, ξ_{2}) = ( \begin{matrix}0 & 1 & 0 \\ −1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) .

The eigenvalues are λ_{1} = −i, λ_{2} = i and λ_{3} = 0, with corresponding eigenvectors

Z_{1} := ( \begin{matrix}i \\ 1 \\ 0 \end{matrix} ) Z_{2} := ( \begin{matrix}−i \\ 1 \\ 0 \end{matrix} ) and Z_{3} := ( \begin{matrix}0 \\ 0 \\ 1 \end{matrix} ) .

Determination of metric from curvature 289

In the associated orthogonal frame (X_{1}, X_{2}, X_{3}) where

X_{1} := ( \begin{matrix}0 \\ 1 \\ 0 \end{matrix} ) X_{2} := ( \begin{matrix}1 \\ 0 \\ 0 \end{matrix} ) and X_{3} := ( \begin{matrix}0 \\ 0 \\ 1 \end{matrix} )

the curvature form is given by

R(∗, ∗) = ( \begin{matrix}0 & ω^{2} ∧ ω^{1} & ω^{2} ∧ ω^{3} \\ ω^{1} ∧ ω^{2} & 0 & ω^{1} ∧ ω^{3} \\ ω^{3} ∧ ω^{2} & ω^{3} ∧ ω^{1} & 0 \end{matrix} )

and so P is satisfied.

Finally, we note that a flat space of dimension greater than one clearly is neither total nor satisfies property P.

References

[1] Atkins, R. Determination of the metric from the connection. arXiv:mathph/0609075.

[2] Atkins, R. When is a connection a metric connection? To appear: New Zealand J. Math.

[3] Hall, G. S. Curvature collineations and the determination of the metric from the curvature in general relativity. Gen. Relativity Gravitation 15 (1983) 581–589. MR0708821 (85c:83023), Zbl 0514.53018.

[4] Hall, G. S.; McIntosh, C. B. G. Algebraic determination of the metric from the curvature in general relativity. International Journal of Theoretical Physics 22 (1983) 469–476. MR0709823 (84k:53057), Zbl 0523.53037.

[5] Ihrig, Edwin. An exact determination of the gravitational potentials g_{ij} in terms of the gravitational fields R_{ijk}^{l}. J. Math. Physics 16 (1975) 54–55. MR0353934 (50 #6416).

[6] Ihrig, E. The uniqueness of g_{ij} in terms of R_{ijk}^{l}. International Journal of Theoretical Physics 14 (1975) 23–35. MR0395712 (52 #16504), Zbl 0323.53021.

[7] McIntosh, C. B. G.; Halford, W. D. Determination of the metric tensor from components of the Riemann tensor. J. Phys A 14 (1981) 2331–2338. MR0628373 (82i:83053), Zbl 0469.53026.

[8] McIntosh, C. B. G.; Halford, W. D. The Riemann tensor, the metric tensor, and curvature collineations in general relativity. Journal of Mathematical Physics 23 (1982) 436–441. MR0644578 (83c:83012), Zbl 0491.53018.

Department of Mathematics, Trinity Western University, 7600 Glover Road, Langley, BC, V2Y 1Y1 Canada richard.atkins@twu.ca

This paper is available via http://nyjm.albany.edu/j/2009/15-15.html.