1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

PERELMAN'S FORMALISM IN POTENTIALLY INFINITE DIMENSIONS 417 PROOF. The sequence ( M^3 ' 9,,.i ( e)' q,,.J' e ::::: 1, is a seq ...

418 8. APPLICATIONS OF THE REDUCED DISTANCE Recall from (7.11) that given a solution (Mn,g(r)), TE (O,T_h of the backward Ricci ...

PERELMAN'S FORMALISM IN POTENTIALLY INFINITE DIMENSIONS 419 1 \laJJj = \laiaj -IIijl/ = \laiaj - ( N)_ 112 Rijl/· R+ 2T By the ...

420 8. APPLICATIONS OF THE REDUCED DISTANCE (R(oi,v)v,aj) = 1 2 [(a7R- 2 N 2 ) Rij + lviRY'jRJ 2 (R+ ~) T (R~ ~) (-a7Rij -t\7i\ ...

PERELMAN'S FORMALISM IN POTENTIALLY INFINITE DIMENSIONS 421 LEMMA 8.40 (O'Neill). Let X, Y, Z be vector fields on B and let U, ...

422 8. APPLICATIONS OF THE REDUCED DISTANCE By (2) we have and hence 1 R(ai, Ba)aj = - JT Hess_g( vr)(ai, aj)Ba, R(ai,Ba)ar ...

PERELMAN'S FORMALISM IN POTENTIALLY INFINITE DIMENSIONS 423 (il(oi,8T)8n8j) = (R(oi,87)8n8j) = 2 (R~~) ((o7R- 2 ~ 2 )~j+t\7iR· ...

424 8. APPLICATIONS OF THE REDUCED DISTANCE Rc(8T) Ba)= gk£ ( R(O-,., 8k)8e, ea)+ ( R(8r, v)v, ea) + ~g^111 ( R(8r, e 1 )e11, Ba ...

PERELMAN'S FORMALISM IN POTENTIALLY INFINITE DIMENSIONS 425 LEMMA 8.41. (1) 900 -m = 900 - - 2of - - -f +^0 (N-1) OT T = N + R ...

426 (2) (3) (4) (5) APPLICATIONS OF THE REDUCED DISTANCE gi()(x,y,T) = ~cp~ (x,y,T) 0 ! 0 (x,y,T)§co(cp(x,y,T)) uX^2 uT o-0 0- ...

PERELMAN'S FORMALISM IN POTENTIALLY INFINITE DIMENSIONS 427 (6) by g~ = ~:~ (x, y, T) °:rd (x, y, T) 9cd (cp (x, y, T)) tr^0 = ...

428 8. APPLICATIONS OF THE REDUCED DISTANCE where V'm is the covariant derivative associated to the metric grJ on M. The other c ...

PERELMAN'S FORMALISM IN POTENTIALLY INFINITE DIMENSIONS 429 Using (6.15), we find that goo= !Joo - J\7fJ 2 = ~ ( ~ - [r(2~f - ...

430 8. APPLICATIONS OF THE REDUCED DISTANCE where I'fj(M, g) and I'~.a(SN) denote the Christoffel symbols of (M, g) and SN, resp ...

PERELMAN'S FORMALISM IN POTENTIALLY INFINITE DIMENSIONS 431 f''Y <Y.(3 -_ 2 1 c-m)'YP({) g <Y.9(3p -m + {) (39Ci.p -m - ...

432 8. APPLICATIONS OF THE REDUCED DISTANCE so that .. n p q A - k£ 8xi 8xJ ~ 81/Jr 81/Jr R = g 8,,Pk 8,,P£ L...J (Rpq + V' P V' ...

Chapter 9. Basic Topology of 3-Manifolds The purpose of this chapter is to introduce certain well-known facts in 3-manifold topo ...

434 9. BASIC TOPOLOGY OF 3-MANIFOLDS one can again obtain a connected sum decomposition M = (5^2 x 5^1 )#M'. This gives more inf ...

INCOMPRESSIBLE SURFACES AND GEOMETRIZATION CONJECTURE 435 was generalized by W. Haken to normal surface theory [1 76].) Let us ...

436 9. BASIC TOPOLOGY OF 3-MANIFOLDS important concept. A compact, connected, properly embedded surface F in a 3-manifold M is s ...