Differential-delay equations with two time lags by Roger D. Nussbaum

By Roger D. Nussbaum

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In both cases M has for r > 0 positive curvature and infinite volume. e. one can have X(M) = X(M, g) as in the finite volume case. For this reason we should find additional conditions which assure in the finite volume case or the infinite volume case, respectively, that 1) X(M, g) is a (proper) homotopy invariant, 2) X(M, g) = X(M) if M has finite topological type. We start with vol( Mn , g) < 00 and IK I ::; 1 where the letter (after rescaling) is equivalent to (Bo). Then X(M,g) = J E(g) M is well defined and for g' E b,2 comp l,2(g) X(M, g) = X(M, g').

D : COO (E) ------t COO(F) be an elliptic operator, (M, g) ------t (M, g) a Riemannian covering, D : C':(E) ------t C':(F) the corresponding lifting and f = Deck (Mn, g) ------t (Mn, g). The actions of f and D commute. If P : L2 (M, E) ------t 1t is the orthogonal projection onto a closed subspace 1t c L 2 (M, E) then one defines the f -dimension dimr 1t of 1t as dimr 1t := trrP, where trr denotes the von Neumann trace and trrP can be any real number ~ 0 or = 00. If one takes 1t = 1t(D) = ker D C L 2 (E), 1t* = 1t(D*) ker(D*) C L2(1') then one defines the f-index indrD as indrD := dimr 1t(D) - dimr 1t(D*).

57]). Let = U +S + W be the corresponding (fiberwise) decomposition into irreducible subspaces. Then this induces for the curvature tensor R = R9 a decomposition R = U + 8 + W. For R = R9 = R+ + R_, we denote by Ric = Ric 9 the Ricci tensor, by 7 = 7 9 the scalar curvature, by K = K9 the sectional curvature and by W = W9 = W+ + W_ the Weyl tensor. There are decompositions for the pointwise norms Ilx as follows 2 2 2 IRI2 IR+12 + IR_12 = IUI + 181 + IWI 41W+12 + IW_12 + 21Ric 12 2 2 61UI + 2181 , 2 241U1 .

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