One of the standard metrics in global coordinates that I found being used for Euclidean AdS is this,
$$ds^2 = \frac{1}{z^2}(dt^2 - dz^2 - \sum_{i=1}^{p-1} dx_i^2)$$
But this is not positive definite unless $p$ is even! So how is this a correct metric that physics papers in particular seem to be using for Euclidean AdS? (I understand that EAdS is defined to be a Riemannian manifold as opposed to just AdS)
Is there an implicit interpretation of it which is different?
Is there a presentation of global metric on Euclidean AdS which is manifestly positive definite?
Some references also use $ds^2 = \frac{1}{z^2}(dz^2 - dt^2 + \sum_{i=1}^{p-1} dx_i^2)$ which tries make the point that AdS is foliated by flat Minkowski spaces. But then its unclear to me why this metric is also claimed to be an ``Euclidean AdS" given that this is also not positive definite!
Any explanations clarifying this would also be helpful!