1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

5. NOTES AND COMMENTARY 181

(1) (Bieberbach theorem) A closed flat n-manifold is isometric to

the quotient of a flat n-torus; see Theorem 3.3.1 and Corollary

3.4.6, both in Wolf [189].

(2) (Splitting and soul theorems) Let (Fn, h) be a complete non-

compact flat n-manifold.

(a) Then a finite cover of p::n, h) is isometric to JRk x 7n-k with

0 < k::; n and where 7n-k is a flat torus.

(b) Moreover, ( :;:n, h) is also isometric to a rank k flat real vector

bundle over a closed flat (n - k)-manifold, where 0 < k::; n.

( c) There is a real analytic deformation retraction of ( :;:n, h) onto

a compact totally geodesic submanifold; see Theorem 3.3.3 of

[189].

For example, we have the Mobius band: F^2 = (5^1 x IR) /'ll 2 , where the

'll2-action is generated by the antipodal map. Then F^2 is nonorientable and

is isometric to a flat line bundle over IRP^1 9'! S^1.

The results in (2) apply to (Fn, h) = ( M~, g 00 (t)) in (21.58). In

particular, a finite cover of (M~, § 00 (t)) is isometric to JRk x 7n-k with

0 < k < n and where 7n-k is a flat torus.

REMARK 21.20. Let r denote the group of deck transformations of the

covering space

7r: l.Rk x rn-k-+ M~.

Let x 00 E JRk x 7n-k be a lift of x 00 , i.e., 7r (x 00 ) = x 00 • Let H 00 denote the

fundamental solution to the adjoint heat equation on (JRk x 7n-k) x (-oo, 0)

based at (x 00 , 0) and let v 00 denote the corresponding Perelman's differential
Harnack quantity (defined analogously to (21.60)). We then have

H 00 (7r (x), t) = L H 00 (/ (x), t)

')'Er

for all x E JRk x 7n-k and t E (-oo, 0) (the sum is finite since Ir! < oo ).

However we do not have the analogous formula relating v 00 and Voo since

they depend nonlinearly on H 00 and H 00 , respectively.