Weyl Tensor Components in n-Dim Manifold: N=3?

[paweld]

I found the formula for the number of independent components of
Weyl tensor in n-dimensional manifold:
$$(N+1)N/2 - \binom{n}{4} - n(n+1)/2~~~~~N=(n-1)n/2$$
This expression implies that in 3 dimension Weyl tensor has 0 independent
components, so it's 0. Does it implies that any three-dimensional manifold
is conformally flat (maybe the formula I've written above is incorrect for n<4)?

 

[bcrowell]

WP has some infromation that seems relevant: http://en.wikipedia.org/wiki/Weyl_tensor

The Weyl tensor has the special property that it is invariant under conformal changes to the metric. That is, if g′ = f g for some positive scalar function f then the (1,3) valent Weyl tensor satisfies W′ = W. For this reason the Weyl tensor is also called the conformal tensor. It follows that a necessary condition for a Riemannian manifold to be conformally flat is that the Weyl tensor vanish. In dimensions ≥ 4 this condition is sufficient as well. In dimension 3 the vanishing of the Cotton tensor is a necessary and sufficient condition for the Riemannian manifold being conformally flat. Any 2-dimensional (smooth) Riemannian manifold is conformally flat, a consequence of the existence of isothermal coordinates.

Physically, I would associate the Weyl tensor with the existence of tidal forces, which conserve the spatial volume of a free-falling cloud of particles. In 2+1 dimensions, this can exist. In 1+1, it can't. From the WP article, it sounds like maybe the Cotton tensor, rather than the Weyl tensor, is the appropriate way to measure tidal forces in 2+1...?

 

[paweld]

Thanks for answer. The vanishing of Weyl tensor is nessesary condition but not sufficient
for a metric to be conformally flat. In dimension >=4 it turns out to be also sufficient
condition but in lower dimensional manifold it is not.
In Wiki article http://en.wikipedia.org/wiki/Weyl_tensor#Conformal_rescaling
about Weyl tensor there is said that metric is conformally flat if it fullfil some PDE
and in dimension >=4 the only integrability condition for this PDE is Weyl tensor=0.
Maybe someone knows what this PDE has in common with conformal flatness.
I have no idea, but it looks interesting.
I'm also looking for simple prove that conformal mapping doesn't change Weyl tensor
(I would love to find geometrical argument, but probably it's imposibble).
Thanks for help

 

[Altabeh]

I'm also looking for simple prove that conformal mapping doesn't change Weyl tensor
(I would love to find geometrical argument, but probably it's imposibble).
Thanks for help

There is no other alternative to the proof of this theorem. Actually it is all clear that a conformal transformation of metric tensor, i.e. $$\bar{g}_{ab}=e^{2\lambda}{g}_{ab},$$ with $$\lambda$$ being a scalar function of coordinates $$x^{a}$$ using a direct calculation, including the expansion of Christoffel symbols and Riemann tensor in the new coordinates, leaves the Weyl tensor unchanged and this all doesn't exceed two ordinary textbook pages.

See e.g. Riemannian Geometry, E. P. Eisenhart, Princeton Press 1949, pp 89-90.

I found the formula for the number of independent components of
Weyl tensor in n-dimensional manifold:

There's already a formula to calculate the number of the independent components of the WT and it is

$$\frac{1}{12}n(n+1)(n+2)(n-3)$$

which gives 0 for n=3 and obviously works fine for n's greater than 2. I don't know how you manage to derive your formula but if it works for any given n, you should first consult an expert and then think of a suitable journal to publish it in (though I for one am sure it has nothing new to tell but as a tacit work can be good.)

This expression implies that in 3 dimension Weyl tensor has 0 independent
components, so it's 0. Does it implies that any three-dimensional manifold
is conformally flat (maybe the formula I've written above is incorrect for n<4)?

Nupe. Any three dimensional pseudo-Riemannian* manifold is conformally flat iff the third order http://en.wikipedia.org/wiki/Cotton_tensor" vanishes.

* (this must be pseudo-Riemannian Weyl tensor is a measure of the pseudo-Riemannian manifolds)

AB

 

[paweld]

Thanks for your answer.
My formula is completely equivalent to yours - I've factorised mine and got the same result.

Maybe you know why in dimension >=4 metric is confomally flat iff Weyl tensor=0

 

[Altabeh]

Thanks for your answer.
My formula is completely equivalent to yours - I've factorised mine and got the same result.

Maybe you know why in dimension >=4 metric is confomally flat iff Weyl tensor=0

See e.g. Riemannian Geometry, E. P. Eisenhart, Princeton Press 1949, pp 91-92.

If you hit anything difficult to catch on, simply ask it here!

AB