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The Einstein Tensor and Its Generalizations
1.Unless otherwise specified, Latin indices run from 1 to n.
2.A comma denotes partial differentiation.
3.The summation convention is used throughout. The vertical bar denotes covariant differentiation.
4.E. Cartan, J. Math. Pure Appl. 1, 141 (1922).
5.H. Weyl, Space‐Time‐Matter (Dover, New York, 1922), 4th ed., pp. 315ff;
5.H. Vermeil, Nachr. Ges. Wiss. Göttingen, 334 (1917).
6.If is any contravariant vector field, then we define the Riemann curvature tensor the Ricci tensor the curvature scalar R, and the Einstein tensor by and , respectively.
7.D. Lovelock, Aequationes Math. 4, 127 (1970).
8.Reference 7, Theorem 4.
9.Reference 7, Theorem 3.
10.Reference 7, Corollary 1.
11.For various applications of (2.6), see D. Lovelock, Atti Accad. Nazi. Lincei 42, 187 (1967);
11.D. Lovelock, Proc. Cambridge Phil. Soc. 68, 345 (1970).
12. Without loss of generality, we may assume
13.Reference 7, Theorem 5.
14.This scalar has arisen elsewhere in an entirely different context. H. Rund, “Curvature Invariants Associated with Sets of n Fundamental Forms of Hypersurfaces of n‐Dimensional Riemannian Manifolds” [to appear in Tensor (1971)].
14.D. Lovelock, “Intrinsic Expressions for Curvatures of Even Order of Hypersurfaces in a Euclidean Space” [to appear in Tensor (1971)].
15.H. Rund, Abhandl. Math. Sem. Univ. Hamburg 29, 243 (1966).
16.The relationship of (3.6) to Lagrangians which satisfy the Euler‐Lagrange equations identically has been investigated by R. Pavelle (private communication).
17.D. Lovelock, Arch. Rati. Mech. Anal. 33, 54 (1969).
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