Jean-Pierre Bourguignon A MATHEMATICIAN'S VISIT TO KALUZA ...

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152where g is a chosen G-invariant metric on the model space of the fibers (necessarilya space where G acts) that can be extended as a bilinear form onthe whole of TM thanks to the connection. Indeed, for a tangent vector Z toM, we decompose it first into its horizontal and vertical components thanksto the connection, say Z = Z v + Zh, and set(13) RZ % Z) = 9(Tv(Z)>Tv(Z)) + HZ %ì Z m ) .The first remarkable fact is then that the G-bundles can be characterizedin terms of metrics like g that we just defined. We come to this a little later.Another fact of concern to us is of course the expression of the LagrangianS for the metric g. Anticipating a bit curvature calculations that we developlater, we have(14) S(5)= / Sv s = I (sop + s-\Q a \ 2 )v dJM JMwhere s,s and 5 denote respectively the scalar curvatures of the metrics g,gand g. In other wordsFACT 4. - The Hilbert-Einstein Lagrangian of General Relativity evaluatedon the extended space-time according to the Kaluza-Klein prescription is thesum of the Hilbert-Einstein Lagrangian of the base and fibers and of theYang-Mills functional of the connection.In the Physics literature, this fact is often referred to as minimal coupling.b) The Riemannian geometry of fibre bundlesA systematic study of the Riemannian geometry of fibre bundles wasconsidered only recently (see [13]) if one compares to the time of existence ofRiemannian geometry itself.In this setting, the basic notion is that of a Riemannian submersion,i.e., a map p : (M,g) —> (M,g) so that, at each point m of M, the restrictionof Trhp to the orthogonal space to its kernel at m is an isometry from themetric induced by g to (Tp^jAf,^^)). To stick to the discussion we startedfrom the Physics side, we will assume that p : (M,g) -*• (M,g) is a fibre bundlewhose fibre is denoted by F. Note however that completeness (a fortioricompactness) and connectedness of (M,g) ensure that property. (This is atheorem of R. Hermann). At this point, it is important to stress that it wouldbe inconsistent to assume that F is endowed with a Riemannian metric. Wecome back to this point later.
The key observation due to B. O'Neill is that two 3-tensor fields areenough to describe the geometry of the situation. They measure the difFerenceof D, the Levi-Civita connection of g, and its projection onto the verticaland horizontal subbundles respectively. If h and v denote the orthogonalprojections onto the horizontal and vertical spaces respectively, for tangentvector Z\,Z 2 to M, we have153(15) / A * lZ *= ^^1)^2)) + v &H*t) h ( z *)){T Zl Z 2 = h(D v(Zl) v(Z 2 )) + v(D v(Zl) h(Z 2 )) .The 3-tensor A measures the obstruction to integrability of this horizontaldistribution. It can be identified with the curvature of the connection.This can be done as follows. The 3-tensor A is defined over M, but one easilysees that A%Y = v([X,Y]), so that A gives rise to a 2-form on M takingits values in vector fields along the fibre. It can therefore be identified withQ°. The horizontal part of A, namely A%V for a vertical vector V, can berecovered from the previous one since g(A^Y,V) = -g(A^V f Y).The 3-tensor T measures the deviation of the fibres from being totallygeodesie. The importance of T is best captured by the following theorem.THEOREM 5 (R.HERMANN, cf. [6]). - If the fibres of a Riemannian submersionp : (M,g) —• (M,
Page 1: Rend. Sem. Mat. Univ. Poi. TorinoFa
Page 4 and 5: 146having the charge density as tim
Page 6 and 7: 148This approach immediately raised
Page 8 and 9: 150the composition of u, viewed as
Page 12 and 13: 154REMARK. - Before going on workin
Page 14 and 15: 156the base. By using (19), orie se
Page 16 and 17: 158the Hopf fibration S 15 -+• Ì
Page 18 and 19: 160dimensionai fibre. Following the
Page 20 and 21: 162We hope to have by now convinced

Jean-Pierre Bourguignon A MATHEMATICIAN'S VISIT TO KALUZA ...

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