Differential Geometry

Download Collected papers of V. K. Patodi by Honorary Professor Michael Atiyah Sir, M S Narasimhan PDF

By Honorary Professor Michael Atiyah Sir, M S Narasimhan

Vijay Kumar Patodi used to be a super Indian mathematicians who made, in the course of his brief lifestyles, primary contributions to the analytic evidence of the index theorem and to the learn of differential geometric invariants of manifolds. This set of accumulated papers edited via Prof M Atiyah and Prof Narasimhan comprises his path-breaking papers at the McKean-Singer conjecture and the analytic facts of Riemann-Roch-Hirzebruch theorem for Kähler manifolds. It additionally includes his celebrated joint papers at the index theorem and the Atiyah-Patodi-Singer invariant.

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1) drar K - ■ K\ ra? a^-^k dZ? £ and and ^ = IpJL , dz. ' , where f Jr denotes the complex conjugate of /"£,. From now on we shall assume that the metric g is a Kaehler metric. The Kaehler property is equivalent to the following relation of summetry: Vr — pr 1 a? — L fa (that is the hermitian connection has no torsion). 29 262 V. K. •■ A r f z J . ) form denned on U with values in the vector boundle £ (thus p e C-(£/, f ® /IT* 0 ''(*))). ftfdZ. * ® J'-'CY) ® T* 0I (Z)). Then for each x e £/, A(x) is an element of £x ® $* T°^(X) ® Tf'KX) and hence defines an endomorphism D«(^W) of f, A T J ' - ' d ) .

R, . • ■ xk in q=0 [ det {x,Ax + ■ • • + xkAk) if k = n . 1 of [3]). Let ,T„ ■ • -,xk be ^-parameters. Then we have det (/ - e*'4' ■ ■ ■ ex*A*) = £ ( - 1 ) " Tr (Aq{eXiAl • • • eXkA*)) 5=0 = t ( - l ) ' T r (e1'0"-1' ■ ■ ■ eXkDD"A> ■ ■ ■ eXkD9A"), we get the result. Let V, W bs complex vector spaces, and n the dimension of V. For (W, WW)xHom(V, O x Horn (F, F)V)-*Hom(W® ->Hom(W® A"V, 0 < < ? < « , let A' W®A"V) ,-l U , , be the map defined by Q (pq(B, C) = B (g> D«C , Be Horn (W, W), C 6 Horn (V, V) .

Let u and

s == ^£ KK^jdZf ^ ^ K AArf2dz, « • . ,) and {K'-) are local expressions for the curvature forms of hermitian connections in the bundles $ and T(X) respectively. 1) ( £ ( - 1)' Tr U""(z'. z')) e = £

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