By Al Ewing

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**Example text**

165) holds for some function λ. 162), we find that 1 2 ∇j R = ∇j λ . 165) with g ij we obtain R = nλ. 168) and hence λ must be a constant if n > 2. Einstein metrics are of considerable importance in physics and mathematics, and we shall encounter them frequently later in the course. Since they are obtained by contracting indcies on the Riemann tensor, the information contained in the Ricci tensor or Ricci scalar is in general less than that contained in the full Riemann tensor; the mapping is non-reversible and one cannot reconstruct the Riemann tensor from the Ricci tensor.

95) on Γ i jk . We shall not give details here, since it would be a bit of a diversion from the main thread of the development. 142) can be written as Ri [jk ] = 0 , ∇[m Ri |j|k ] = 0 . 143) In writing the Bianchi identity in this way we have introduced another piece of standard notation, namely that indices enclosed by vertical lines, such as |j| in the above, are omitted from the antisymmetrisation. The Riemann tensor characterises the curvature of the metric g ij . To see how this works, first let us consider the case of flat Euclidean space, with the metric g ij = δij .

110) This is known as the Christoffel Connection, or sometimes the Affine Connection. 99) under general coordinate transformations. Actually, there is really no need to check this point, since it is logically guaranteed from the way we constructed it that it must have this property. So we leave it as an “exercise to the reader,” to verify by direct computation. 110) to calculate Γ i jk , in terms of ∂i and gij (which can be expressed in terms of ∂ i and gij using their standard tensorial transformation properties).

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