Existence results of Spin$(2,n-1{)}_0$-pseudo-Riemannian cobordisms

In this note, we study necessary and sufficient conditions for the existence of a Spin $(n+1)$-dimensional cobordism that supports a non-singular and non-degenerate pseudo-Riemannian metric of signature $(2,n-1)$, which restricts to a non-singular time-orientable Lorentzian metric on its boundary. The corresponding cobordism groups are computed.