By Honorary Professor Michael Atiyah Sir, M S Narasimhan

Vijay Kumar Patodi was once a super Indian mathematicians who made, in the course of his brief existence, basic contributions to the analytic facts of the index theorem and to the research of differential geometric invariants of manifolds. This set of accumulated papers edited by way of Prof M Atiyah and Prof Narasimhan contains 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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**Extra info for Collected papers of V. K. Patodi**

**Example text**

A - d\B%{t,z',z) The integer N is to be chosen larger than n, and these conditions determine the double forms Ui,9(z',z) uniquely in a sufficiently small neighbourhood of the diagonal as we shall see now. We have 4xp(-ir)) - g? a dz? {< + (dzr\dsy} 4 = ±

Moreover for any positive integer N we have the following Minukshisundaram expansion for ep(t,x,x,); (See § 4 of [7]). Jx) + ... -;ht„p IOis ial «C°°-function on M. x) = ( d\ (4w«)-«* X (volumeof M) + 0(< l ~* 2 ). (;') < - " - (3) tho for any fixed p, 0 < p < d , the volume From PYoi (1) and (3) itJ follows tliat and the dimension of the manifold can be heard from the spectrum {An„}. Let aiT = (volume of M)~l x | k^x), i> 1, 0 < p< d. „ + ... i'). U o. 0. P. M. Singer [4] have raised the following ques tion.

2 + /, + / < / ! Let X, be C^-sections of £®£*® 7*'(AT)® T'*"A(X) defined over U and fl„ ■ ■ •, Bh be arbitrary elements of C°°{U, f ® f *). Let a be a permutation of {I, ■ ■ ■, /, + /3}. For any integer q, 0 < q < n, define the endomorphisms S'j. <8> AT*«''(I)} by lD"(A{), for 1 < i < U , S'o, = \Bt_u ® /„, for / , < / < / , + /3, /„ bemg the identity endomorphism 1 o/AT*w(I). Lef f/ie operators VXl, ■ ■ ■, VXh be denoted by F,, ■ ■ •, Vh. Then we have t ( - D" Tr [5? o . „+;a o F , o .