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By Norman Steenrod

Fibre bundles, now a vital part of differential geometry, also are of significant significance in smooth physics--such as in gauge thought. This publication, a succinct advent to the topic by means of renown mathematician Norman Steenrod, used to be the 1st to provide the topic systematically.

It starts off with a common advent to bundles, together with such themes as differentiable manifolds and overlaying areas. the writer then presents short surveys of complex issues, akin to homotopy conception and cohomology conception, ahead of utilizing them to check extra houses of fibre bundles. the result's a vintage and undying paintings of serious application that might attract critical mathematicians and theoretical physicists alike.

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3. E(S) harmonic submanifold = E inner The energy 7. M is End(V) I S2 Fnd(EV) I XS -SX}, I YS SY}. JY E End(fff) IX (indefinite) < defined = fm := functional < dS A *dS >. with respect to variations of S are called M to Z. S is harmonic if and only if the Z-tangential component of dS vanishes: (d This condition is equivalent to * any dS)T of the F. E. : LNM 1772, pp. 31 - 38, 2002 © Springer-Verlag Berlin Heidelberg 2002 = 0. 2) = 0, (6-3) = 0. 4) fact, d(S Proof. St be Let a Q * = 4d * of S in variation fm d d Wt- E(S) the 4d = S(d dS)T * Z with = (Sd variational * dS)T.

HIE E is is the called a is equivalent to the fact 0 E F(E), projection. 1) do) and hence induces a holomorphic structure F(E) into itself, complex line bundle E. line bundle A complex line bundle E C H induces a quaternionic of H maps the on L=EH=EE)Ej CH. to the structure of a JE admits a unique extension complex structure bundle (L, J), namely right-multiplication complex quaternionic by (-i) on line bundle (L, J) C H induces a a complex quaternionic Ej. Conversely, line bundle complex The E:= Definition lift 10.

HP1 with deriva- L C H in curve Then there exist complex structures J such that We want to extend J and j = to S E a jj complex structure F(End(H)) such that SL this Q case". e. 4) dSL c L. of S is'clear: The existence bundle L' Since L' Identify not unique, L. is H = 0, R can be Q: interpreted I((S 4 + 1(SdS 4 Q+ V) E F(L), SIL complementary Ji SIP := j. := S + R is kerR, and We compute If some if and RS + SR Note that for and define 7r, RH c L c whence R2 L E) L' = It is easy to see that S is not unique.

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