Ricci - flat warped product

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Description:

• Einstein manifolds (manifolds with constant Ricci curvature) are riemannian manifolds with metric tensor field  and Ricci tensor field  where .(see Arthur L. Besse. "Einstein Manifolds"  Springer-Verlag). Thus for Ricci flat manifolds we have  and   .
• Warped products is a simple class of riemannian submersions which is defined as follows. Let  be riemannian manifold  (base space) with metric  and  be riemannian manifold  (fiber space) with metric . Then riemannian manifold  (total space) where  is positive function on  is warped product with warped function .
• In this worksheet we deal with warped product with 2-dimensional base  and metric:
    where  is complete metric on p-dimensional Einstein manifold  with Ricci constant  and . We take p-dimensional sphere  as the fiber space .

restart:
with(atlas):

Total space - M

Domain(M);

Constants:
Constants(Lambda);

Vector fields:
Vectors(E[i],X,Y,Z);

Differential p-forms:
Forms(e[j]=1);

p-Sphere dimension (change it here):
p:=2;

Coframe 1-forms:
Coframe(e[1]=d(rho),e[2]=d(theta),seq(e[i]=d(x[i-2]),i=3..p+2));

Frame vector fields:
Frame(E[i]);

Metric tensor fie ld:
Metric( g=1/(1-rho^(1-p))*d(rho)&.d(rho)+4*(1-rho^(1-p))/(p-1)^2*d(theta)&.d(theta)+rho^2*4*add(d(x[i])&.d(x[i]),i=1..p)/(1+add(x[i]*x[i],i=1..p))^2);

Connection 1-forms:
Connection(omega);

Curvature 2-forms:
Curvature(Omega);

Curvature tensor field:
Riemann(R);

Verify that total space is Ricci flat:
Ricci(r);

Base space - B

Declare base domain:
Domain(B);

Declare forms:
Forms(u[i]=1);

Declare vectors:
Vectors(U[k]);

Declare coframe
Coframe(u[1]=d(zeta),u[2]=d(xi));

Frame declaration:
Frame(U[k]);

Let us define metric on the base:
Metric(G=1/(1-zeta^(1-p))*d(zeta)&.d(zeta)+4*(1-zeta^(1-p))/(p-1)^2*d(xi)&.d(xi));

Submersion definition

Let us define submersion   :  such that   :
Mapping(pi,M,B,
        zeta=rho,
        xi=theta);

Who(pi);
pi: mapping

Projectors of the submersion

Now we can calculate vertical projector V  and horizontal projector H :
P:=Projectors(pi);

V:=P[vertical];

H:=P[horizontal];

Thus vertical and horizontal projections of arbitrary vector X are:

'iota[X](V)'=iota[X](V);
'iota[X](H)'=iota[X](H);

Invariants T and A of the submersion:

Let us calculate invariants of the submersion:

Inv:=Invariants(pi);

So, submersion invariant  is equal to zero. Thus obstruction against integrability of the horizontal distribution is equal to zero. It is obvious that the submersion is a riemannian one but we can verify it directly. To do this we "rise" G metric into total space using restriction operator `&/`:

G &/ pi;

We obtain the horizontal part of g metric.

For tensor field  we have:

T:=eval(Inv['T']);

To construct the T - tensor:

TT:=add(add(`&.`(e[i],e[j],T[i,j]),j=1..p+2),i=1..p+2);

For vectors X and Y:

'T[X](Y)'=iota[X,Y](TT);

Jump to total manifold:

Domain(M);

Now for coordinate representation of T we obtain:

'iota[E[i],E[j]](T)'=iota[E[i],E[j]](TT);

For mean curvature vector field  where  is vertical projection of the metric tensor  we obtain:

N:=eval(Inv[meanCurvature]);

But for warped product we have  , where  is warped function. Let us verify that:

'N'=-p/rho*grad(rho);