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,y1f1(x1,x2...xm),y2f2(x1,x2...xm),...,
ynfn(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:
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Total space - M

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Declare some constants:
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Vector fields:
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Differential p-forms:
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p-Sphere dimension (change it here):
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Coframe 1-forms:
Frame vector fields:
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Metric tensor field:
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Connection 1-forms:
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Curvature 2-forms:
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Curvature tensor field:
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Verify that total space is Ricci flat:
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Base space - B

Declare base domain:
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Declare forms:
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Declare vectors:
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Declare coframe:
Frame declaration:
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Let us define metric on the base:
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Submersion definition

Let us define submersion :MB such that =, =:
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Projectors of the submersion

Now we can calculate vertical projector V and horizontal projector H:
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Thus vertical and horizontal projections of arbitrary vector X are:
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Invariants T and A of the submersion:

Let us calculate invariants of the submersion:
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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 `&/`:
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We obtain the horizontal part of g metric. For tensor field Tx(Y)=HDVX(VY)+VDVX(HY) we have:
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To construct the T - tensor:
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For vectors X and Y:
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Jump to total manifold:
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Now for coordinate representation of T we obtain:
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For mean curvature vector field K=traceVg(T) where Vg is vertical projection of the metric tensor g we obtain:
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But for warped product we have , where f=2 is warped function. Let us verify that:
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