My question is:


Can we judge a manifold that can admit a (p,q) metric? 


I only know the case that the existen of a lorentz metric is equivalent to Euler Character is zero



A Existence Problem of (p,q) metric
The criterion for existence of a $(p,q)$ metric is (assuming $p+q=dim X$) that the tangent bundle splits as a direct sum of two subbundles of dimensions $p$ and $q$. 


EDIT : I doubt there is an easy algebraic topology criterion in general, as characteristic classes [EDIT after Lennart Meier's comment: other than Euler class, which cannot help if $p,q>1$] are *stable* invariants of vector bundles. At least, one has necessary conditions, such as : the total Stiefel-Whitney and Ponttryagin (Chern of complexified bundle) class of $TX$ is the product of two (inhomogeneous) classes of degrees at most $p$ and $q$. Perhaps real $K$-theory (and its operations) gives better necessary conditions (although still not sufficient in general). But it's not easy to compute.


