Weyl tensor in 2 dimensions- confused

[zn5252]

hello,
The Weyl tensor is:

http://ars.els-cdn.com/content/image/1-s2.0-S0550321305002828-si53.gif

In 2 dimensions , the Riemann tensor is (see MTW ex 14.2):
R_abcd = K( g_acg_bd - g_adg_bc ) [R]

Now the Weyl tensor must vanish in 2 dimensions. However, working with the g

g =
[-1 0 0 1 ]
[ 0 1 0 0 ]
[ 0 0 1 0 ]
[ 0 0 0 1 ]

Will yield a null Weyl tensor indeed with a Ricci scalar of 12K (from formula R). (see solution of Pb 9.27 in Lightmann)
But in 2 dimensions with the g :

g =
[-1 0 ]
[ 0 1 ]

Will not give a null Weyl tensor ! with a Ricci scalar 2K (from formula R).

Now I'm confused. The issue is very easy though.
Where did I go wrong ?

 

[zn5252]

All N = 2 spaces are conformally flat.
This would mean that since the Weyl tensor vanishes for the conformal space whose Riemann tensor has the form [R], thus one can conclude that for N=2, the Weyl tensor is null.
This might make sense. But i do not know why the computation above did not yield a null Weyl tensor. For N=2 , the Weyl tensor is undefined, thus the link that I provided is not valid for N=2.

 

[zn5252]

hello,
The Weyl tensor is:

http://ars.els-cdn.com/content/image/1-s2.0-S0550321305002828-si53.gif

In 2 dimensions , the Riemann tensor is (see MTW ex 14.2):
R_abcd = K( g_acg_bd - g_adg_bc ) [R]

Now the Weyl tensor must vanish in 2 dimensions. However, working with the g

g =
[-1 0 0 1 ]
[ 0 1 0 0 ]
[ 0 0 1 0 ]
[ 0 0 0 1 ]

Will yield a null Weyl tensor indeed with a Ricci scalar of 12K (from formula R). (see solution of Pb 9.27 in Lightmann)
But in 2 dimensions with the g :

g =
[-1 0 ]
[ 0 1 ]

Will not give a null Weyl tensor ! with a Ricci scalar 2K (from formula R).

Now I'm confused. The issue is very easy though.
Where did I go wrong ?

This is related to the last question of MTW ex 21.21