3.7 The Levi Civita Connection |
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Fig. 3.18 James Joseph Sylvester (1814–1897) was an eminent English Mathematician who gave fundamental contributions in matrix theory, group theory and number theory. Of Jewish origins he had to suffer several discriminations in the course of his career and also had to law-suite the Royal Military Academy that refused to pay his full-pension. He crossed twice the Atlantic Ocean to become an American Professor. Once at University of Virginia in 1843 where he stayed only six months and a second time in 1877 when he was appointed inaugural professor of mathematics at the newly founded John Hopkins University of Maryland. In the USA he founded the American Journal of Mathematics. Towards the end of his long life he returned to England and finally received several honors from the Royal Society that in 1901, after his death, instituted the Sylvester Medal in his memory
(p, m − p), which is intrinsic to the matrix A. This is what happens for a single matrix. Consider now a point dependent matrix like the metric tensor g, whose entries are smooth functions. Defining s = 2p − m the difference between the number of positive and negative eigenvalues of g it follows that also s is a smooth function. Yet s is an integer by definition. Hence it has to be a constant.
The metrics on a differentiable manifold are therefore intrinsically characterized by their signatures. Riemannian are the positive definite metrics with signature (m, 0). Lorentzian are the metrics with signature (1, m − 1) just as the flat metric of Minkowski space. There are also metrics with more elaborate signatures which appear in certain mathematical problems although they are not immediately relevant for General Relativity.
3.7 The Levi Civita Connection
Having established the rigorous mathematical notion of both a metric and a connection we come back to the ideas of Riemannian curvature and Torsion which were
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3 Connections and Metrics |
heuristically touched upon in the course of our historical outline. In particular we are now in a position to derive from clear-cut mathematical principles the Christoffel symbols anticipated in (3.2.7), the Riemann tensor mentioned in (3.2.10) and the Torsion tensor sketched in (3.2.13). The starting point for the implementation of this plan is provided by a careful consideration of the special properties of affine connections.
3.7.1 Affine Connections
In Definitions 3.4.1, 3.4.2 we fixed the notion of a connection on a generic vector
π π
bundle E = M . In particular we can consider the tangent bundle T M = M . A connection on T M is named affine. It follows that we can give the following:
Definition 3.7.1 Let M be an m-dimensional differentiable manifold, an affine connection on M is a map
: X(M ) × X(M ) → X(M )
which satisfies the following properties:
(i)X, Y, Z X(M ) : X(Y + Z) = XY + XZ
(ii)X, Y, Z X(M ) : (X+Y)Z = XZ + YZ
(iii)X, Y X(M ), f C∞(M ) : f XY = f XY
(iv)X, Y X(M ), f C∞(M ) : X(f Y) = X[f ]Y + f XY
Clearly also affine connections are encoded into corresponding connection oneforms, which are traditionally denoted by the symbol Γ . In the affine case Γ is gl(m, R)-Lie algebra valued since the structural group of T M is GL(m, R). Let {eμ} be a basis of sections of the tangent bundle so that any vector field X X(M ) can be written as follows:
X = Xμ(x)eμ |
(3.7.1) |
The connection one-form is defined by calculating the covariant differentials of the basis elements:
eν = Γν ρ eρ |
(3.7.2) |
Introduce the dual basis of T M , namely the set of one-forms ωμ such that:
ωμ(eν ) = δνμ |
(3.7.3) |
The matrix-valued one-form Γ can be expanded along such a basis obtaining:
Γ = ωμΓμ Γν ρ = ωμΓμν ρ |
(3.7.4) |