TOPOLOGY AND GEOMETRY IN PHYSICS
1 LET M BE A MANIFOLD, LET G BE A LIE GROUP AND LET PRM : M × G ?M BE THE NATURAL PROJECTION THEN, (M × G, G, M, ?, PRM), WITH ? GIVEN BY RIGHT TRANSLATION OF G ON THE SECOND FACTOR OF M × G, IS A PRINCIPAL BUNDLE, CALLED THE PRODUCT PRINCIPAL BUNDLE IT IS OBVIOUSLY TRIVIAL 2 LET (P, G, M, ?, ?) BE A PRINCIPAL BUNDLE AND LET U ?M BE OPEN DE?NE PU := ??1(U) AND TAKE THE RESTRICTIONS ?U : PU ?U OF ? AND ?U : PU × G ?PU OF ? BY INTERSECTION, ANY LOCAL TRIVIALIZATION OF P INDUCES A LOCAL TRIVIALIZATION OF PU THUS, (PU, G, U, ?U, ?U) IS A PRINCIPAL G-BUNDLE OVER U TOPOLOGY AND GEOMETRY IN PHYSICS ? ?? G = ? 2THIS STATEMENT ALSO FOLLOWS FROM THEOREM I/5 7 2 AND REMARK I/5 7 3 TOPOLOGY AND GEOMETRY IN PHYSICS 1 (1 1 4) 2 IF THE PRINCIPAL BUNDLES P AND Q HAVE THE SAME BASE MANIFOLD M AND IF ˜? = IDM, THEN (?, ?) IS SAID TO BE VERTICAL IF P AND Q HAVE THE SAME STRUCTURE GROUP G AND IF ? = IDG, THEN ? IS CALLED A G-MORPHISM BY LOCAL TRIVIALITY, EVERY VERTICAL G-MORPHISM IS A DIFFEOMORPHISM AND HENCE AN ISOMORPHISM 3 IF ˜? AND ? ARE INJECTIVE IMMERSIONS, THEN ? IS AN INJECTIVE IMMERSION, TOO IN THIS CASE, P1 IS CALLED A SUBBUNDLE OF P2 (A) IF, ADDITIONALLY, ˜? AND ? ARE EMBEDDINGS, THEN P IS CALLED AN EMBEDDED SUBBUNDLE IN THIS CASE, P1 MAY BE IDENTI?ED WITH THE IMAGE OF THE MORPHISM ? IN P2 (B) IF, ADDITIONALLY, M1 = M2 = M AND ˜? = IDM, THEN P1 IS REFERRED TO AS A ?- REDUCTION OR, SIMPLY, A REDUCTION OF P2 IN THIS CASE, ONE SAYS THAT G1 IS A REDUCTION OF THE STRUCTURE GROUP G2 TWO REDUCTIONS ARE SAID TO BE EQUIVALENT IF THEY DIFFER BY A VERTICAL AUTOMORPHISM OF P ? REMARK 1 1 9 (PULLBACK OF PRINCIPAL BUNDLES) 1 IN COMPLETE ANALOGY TO VECTOR BUNDLES, SEE SECT 2 6 OF PART I, GIVEN A PRINCIPAL G-BUNDLE P OVER M WITH CANONICAL PROJECTION ?, WE DE?NE ITS PULLBACK BY A SMOOTH MAPPING ? : N ?M: ??P := {(Y, P) ?N × P: ?(Y) = ?(P)} THIS IS A PRINCIPAL G-BUNDLE OVER N AND THE CANONICAL PROJECTION N × P ?P RESTRICTS TO A MORPHISM ??P ?P COVERING ? ONE CAN SHOW THE FOLLOWING (EXER- CISE 1 1 4) (A) IF P IS VERTICALLY ISOMORPHIC TO SOME PRINCIPAL G-BUNDLE Q OVER M, THEN F ?P IS VERTICALLY ISOMORPHIC TO F ?Q (B) IF ? : K ?N IS A FURTHER SMOOTH MAPPING, THEN ??(??P) IS VERTICALLY ISO- MORPHIC TO (? ??)?P (C) LET ? : P ?Q BE A PRINCIPAL G-BUNDLE MORPHISM COVERING ˜? : M ?N THE INDUCED MAPPING P ?˜??Q, P ? ?(P), ?(P) , IS A VERTICAL ISOMORPHISM AND ? DECOMPOSES INTO THE COMPOSITION OF THIS ISOMORPHISM WITH THE NATURAL PRINCIPAL G-BUNDLE MORPHISM ˜??Q ?Q 2 THE FOLLOWING IS AN IMPORTANT SPECIAL CLASS OF PULLBACK BUNDLES LET P1(M, G1) AND P2(M, G2) BE PRINCIPAL BUNDLES BY EXAMPLE 1 1 4/3, P1 × P2 CARRIES THE TOPOLOGY AND GEOMETRY IN PHYSICS