By Honorary Professor Michael Atiyah Sir, M S Narasimhan

ISBN-10: 9810226594

ISBN-13: 9789810226596

Vijay Kumar Patodi was once an excellent Indian mathematicians who made, in the course of his brief lifestyles, basic contributions to the analytic facts of the index theorem and to the research of differential geometric invariants of manifolds. This set of gathered papers edited by means of Prof M Atiyah and Prof Narasimhan comprises his path-breaking papers at the McKean-Singer conjecture and the analytic evidence of Riemann-Roch-Hirzebruch theorem for Kähler manifolds. It additionally comprises his celebrated joint papers at the index theorem and the Atiyah-Patodi-Singer invariant.

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For any two vector fields X. Y defined on U. K(X, Y) is an endomorphism of C"(U,T*r(X)) and maps C (U, T*°\X)) into itself. 4) (Fx c FY - r,- = Fx) a = (D"(/; ® K(Y, X)) - S(Y,X)®Ill + Fix,n)(a) , rxeC'(U. £ ® A«T*o'(X)) , L denoting identity endomorphism of | and /,, the identity endomorphism of A'P'(i). 4) for q = 1 is just the definition of curvature tensor. 4) is true for a = T**-l(X)) ,

Seeley. Complex powers of an elliptic operator, Proc. Sympos. Pure Math. Vol. 10, Amer. Math. Soc, 1967, 288-307. [ 5 ] E. Vesentini. Lectures on Levi convexity of complex manifolds and cohomology vanishing theorems, Tata Institute of Fundamental Research. Bombay, 1967. TATA I N S T I T U T E F U N D A M E N T A L RESEARCH, BOMBAY 51 Journal of the Indian Math. Soc. 34 (1970) 269-285 CURVATURE AND THE FUNDAMENTAL SOLUTION OF THE HEAT OPERATOR By V. K . P A T O D I [Received December 10, 1970] 1.

Here we shall not go into the details of the proof which are quite straight forward. For any two vector fields X. Y defined on U. K(X, Y) is an endomorphism of C"(U,T*r(X)) and maps C (U, T*°\X)) into itself. 4) (Fx c FY - r,- = Fx) a = (D"(/; ® K(Y, X)) - S(Y,X)®Ill + Fix,n)(a) , rxeC'(U. £ ® A«T*o'(X)) , L denoting identity endomorphism of | and /,, the identity endomorphism of A'P'(i). 4) for q = 1 is just the definition of curvature tensor. 4) is true for a = T**-l(X)) ,

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


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