Ricci - flat warped product 

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 are defined as follows.
  Let be riemannian manifold (base space) with metric and be riemannian manifold (fiber space) with metric .
  The warped product is the riemannian manifold  where is positive function (warped function) on

 

• 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);

 

(2.1.1)

 

Constants:
Constants(Lambda); 

(2.1.2)

 

Vector fields:
Vectors(E[i],X,Y,Z); 

(2.1.3)

 

Differential p-forms:
Forms(e[j]=1); 

(2.1.4)

 

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

(2.1.5)

 

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

(2.1.6)

 

Frame vector fields:
Frame(E[i]); 

(2.1.7)

 

Metric tensor field:
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); 

(2.1.8)

 

Connection 1-forms:
Connection(omega); 

(2.1.9)

 

Curvature 2-forms:
Curvature(Omega); 

(2.1.10)

 

Curvature tensor field:
Riemann(R); 

(2.1.11)

 

Verify that total space is Ricci flat: 

>

Ricci(r);

 

(2.1.12)

 

Base space - B 

Declare base domain:
Domain(B); 

(2.2.1)

 

Declare forms:
Forms(u[i]=1); 

(2.2.2)

 

Declare vectors:
Vectors(U[k]); 

(2.2.3)

 

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

(2.2.4)

 

Frame declaration:
Frame(U[k]); 

(2.2.5)

 

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));

 

(2.2.6)

 

Submersion definition 

Let us define submersion  : such that  : 

>	Mapping(pi,M,B,        zeta=rho,        xi=theta);

 

 

	(2.3.1)

 

>

Who(pi);

 

 

pi: mapping
	(2.3.2)

 

Projectors of the submersion 

Now we can calculate vertical projector V and horizontal projector H: 

>	P:=Projectors(pi);

 

(2.4.1)

 

>	V:=P[vertical];

 

(2.4.2)

 

>	H:=P[horizontal];

 

(2.4.3)

 

Thus vertical and horizontal projections of arbitrary vector X are: 

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

 

 

	(2.4.4)

 

Invariants T and A of the submersion: 

Let us calculate invariants of the submersion: 

>	Inv:=Invariants(pi);

 

(2.5.1)

 

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;

 

(2.5.2)

 

We obtain the horizontal part of g metric. 

For tensor field we have: 

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

 

(2.5.3)

 

To construct the T - tensor:  

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

 

(2.5.4)

 

For vectors X and Y: 

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

 

(2.5.5)

 

Jump to total manifold: 

>	Domain(M);

 

(2.5.6)

 

Now for coordinate representation of T we obtain: 

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

 

(2.5.7)

 

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

>	N:=eval(Inv[meanCurvature]);

 

(2.5.8)

 

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

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

 

(2.5.9)

 

>