Ricci - flat warped product
Einstein manifolds (manifolds with constant Ricci curvature) are riemannian manifolds with metric tensor field
g and Ricci tensor field
r=
g where
=const (
see Arthur L. Besse. "Einstein Manifolds" Springer-Verlag). Thus for Ricci flat manifolds we have
=0 and
r=0.
Warped products is a simple class of riemannian submersions which are defined as follows. Let
{B, gB} be riemannian manifold
B (base space) with metric
gB and
{F, gF} be riemannian manifold
F (fiber space) with metric
gF.
The warped product is the riemannian manifold
{B×F, gB+fgF} where
f=f(b) is positive function (warped function) on
B.
What we do?
In this notebook we deal with warped product with 2-dimensional base
B=R2 and metric:
where
gF is complete metric on p-dimensional Einstein manifold
F with Ricci constant
F=p-1 and
p=dim(F). We take p-dimensional sphere
{Sp, gcan} as the fiber space
{F, gf}.
Mapping[f,m,n,y1→f1(x1,x2...xm),y2→f2(x1,x2...xm),...,
yn→fn(x1,x2...xm)] | f - variable - the mapping identifier i.e. f : m  n,
m - variable - first domain identifier,
n - variable - second domain identifier. |
| Who[l] | l -an identifier, list or set of identifies. |
| Riemann[id] | id - variable - corresponding identifier. |
| Ricci[id] | id - variable - corresponding identifier. |
Necessary functions.
R2 
Sp
First of all we load Atlas package:
Total space - M
| Out[8]= |  |
| Out[9]= |  |
| Out[10]= |  |
| Out[11]= |  |
p-Sphere dimension (change it here):
| Out[12]= |  |
| Out[14]= |  |
| Out[15]= |  |
| Out[16]= |  |
| Out[17]= |  |
| Out[18]= |  |
Verify that total space is Ricci flat:
| Out[19]= |  |
Base space - B
| Out[20]= |  |
| Out[21]= |  |
| Out[22]= |  |
| Out[24]= |  |
Let us define metric on the base:
| Out[25]= |  |
Submersion definition
Out[27]//TableForm= |
| |  |
Projectors of the submersion
Now we can calculate vertical projector
V and horizontal projector
H:
| Out[28]= |  |
| Out[29]= |  |
| Out[30]= |  |
Thus vertical and horizontal projections of arbitrary vector X are:
| Out[31]= |  |
| Out[32]= |  |
Invariants T and A of the submersion:
Let us calculate invariants of the submersion:
| Out[33]= |  |
So, submersion invariant
Ax(Y)=HDHX(VY)+VDHX(HY) 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 `&/`:
| Out[34]= |  |
We obtain the horizontal part of g metric. For tensor field
Tx(Y)=HDVX(VY)+VDVX(HY) we have:
| Out[35]= |  |
To construct the T - tensor:
| Out[36]= |  |
| Out[37]= |  |
| Out[38]= |  |
Now for coordinate representation of T we obtain:
| Out[39]= |  |
For mean curvature vector field
K=traceVg(T) where
Vg is vertical projection of the metric tensor
g we obtain:
| Out[40]= |  |
But for warped product we have
, where
f=
2 is warped function. Let us verify that:
| Out[41]= |  |
- Lie derivative
- Exterior product
- Tensor product
- Hodge operator
- Covariant derivative
:M
=
=
:
