All tensors of contravariant valency two, which are divergence free on one index and which are concomitants of the metric tensor, together with its first two derivatives, are constructed in the four‐dimensional case. The Einstein and metric tensors are the only possibilities.

1.
Unless otherwise specified, Latin indices run from 1 to n. A comma denotes partial differentiation.
2.
The summation convention is used throughout. The vertical bar denotes covariant differentiation.
3.
E.
Cartan
,
J. Math. Pure Appl.
1
,
141
(
1922
).
4.
H. Weyl, Space‐Time‐Matter (Dover, New York, 1922), 4th ed., pp. 315 ff.;
H.
Vermeil
,
Nachr. Ges. Wiss. Göttingen
,
334
(
1917
).
5.
If Xi is any contravariant vector field, then we define the Riemann curvature tensor Rhjki, the Ricci tensor Rhj, the curvature scalar R, and the Einstein tensor Gij by
X|jki−X|kji = RhjkiXh,
Rhj = Rhjii respectively.
6.
D.
Lovelock
,
J. Math. Phys.
12
,
498
(
1971
).
7.
For applications of this result see
D.
Lovelock
,
Atti Accad. Naz. Lincei Rend.
42
,
187
(
1967
);
D.
Lovelock
,
Proc. Cambridge Phil. Soc.
68
,
345
(
1970
);
D.
Lovelock
,
Matrix Tensor Quart.
21
,
84
(
1971
).
8.
H.
Rund
,
Abhandl. Math. Sem. Univ. Hamburg
29
,
243
(
1966
);
J. C.
du Plessis
,
Tensor
20
,
347
(
1969
).
9.
D.
Lovelock
,
Arch. Ratl. Mech. Anal.
33
,
54
(
1969
).
10.
M. A. McKiernan, “Tensor Concomitants of the Metric Tensor,” Demonstratio Mathematica (to be published).
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