1547671870-The_Ricci_Flow__Chow

8 1. THE RICCI FLOW OF SPECIAL GEOMETRIES

(a) M^3 is a Lie group isomorphic to SU (2);

(b) M^3 is a Lie group isomorphic to ~^3 ; or

( c) M^3 is diffeomorphic to hyperbolic 3-space H^3.

(2) g* ~ SO (2), in which case either

(a) M^3 is a trivial bundle over a 2-dimensional maximal model,

so that

(i) M^3 is diffeomorphic to S^2 x ~; or

(ii) M^3 is diffeomorphic to 'H^2 x ~;

(b) M^3 constitutes a nontrivial bundle over a 2-dimensional max-

imal model, so that

(i) M^3 is a Lie group isomorphic to nil; or

(ii) M^3 is a Lie group isomorphic to SL ---------(2, ~).

(3) g* is trivial, in which case M^3 is a Lie group isomorphic to sol.

4. Analyzing the Ricci flow of homogeneous geometries

In a homogeneous geometry, every point looks the same as any other. As

a consequence, the Ricci fl.ow is reduced from a system of partial differential

equations to a system of ordinary differential equations. In this section, we

shall outline how this reduction may be done for the five model geometries

that can be realized as a pair (9, 9), where g is a simply-connected unimod-

ular Lie group. The theory in these cases is most elegant, but the remaining

three model geometries can be treated in a similar fashion.

Let gn be any Lie group, and let g be the Lie algebra of all left-invariant

vector fields on Q. Since a left-invariant metric on g is equivalent to a scalar

product on g, the set of all such metrics can be identified with the set S;t

of symmetric positive-definite n x n matrices. s;t is an open convex subset

of ~n(n+l)/^2. For each left-invariant metric g on 9, the Ricci fl.ow may thus

be regarded as a path t f----+ g ( t) E S;t. One would thus expect the Ricci fl.ow

to reduce to a system of n (n + 1) /2 coupled ordinary differential equations.

In fact, we can do better.

Let us equip g with a left-invariant moving frame {Fi}. Then the struc-

ture constants c~j for the Lie algebra g of left-invariant vector fields on g

are defined by

[Fi, Fj] = ctFki

and the adjoint representation of g is the map ad : g ----t gl (g) ~ gl (n, ~)

given by

(ad V) W ~ [V, W] = ViWjc~jFk·

If g is a left-invariant metric on g, then the adjoint ad* : g ----t gl (g) with

respect tog of the map ad : g ----t gl(g) is given by

((adX)*Y,Z) = (Y,(adX)Z) = (Y,[X,Z]).

Now suppose g is a 3-dimensional Lie group. Let {Fi} be a left-invariant

moving frame chosen to be orthonormal with respect to a given left-invariant