# Meaning of $n+1$ dimensions

Is there anything mathematically significant about studying a theory in $$n$$ dimensions or $$m+1$$ dimensions if $$n=m+1$$? For instance in the context of general relativity I hear people talk about the theory in 4 dimensions of (3+1) dimensions commonly and I am not sure about what the difference is. As far as mathematical structure is concerned, is it fair to say that the manifold background is diffeomorphic or otherwise structurally identical to a product manifold when the dimensions are written in split notation?

When people say $$3+1$$ dimensions, or $$n+1$$ dimensions, what they mean is that the metric has $$n$$ space dimensions and 1 time dimension. What this means in GR is that the space is locally diffeomorphic to $$n$$-dimensional Minkowski space, with metric $$\text{diag}(1,-1, \ldots -1)$$ (or $$\text{diag}(-1,1, \ldots 1)$$, depending on convention).
• A spacetime isn't necessarily foliable into spacelike hypersurfaces but the term $3+1$ is still used – Slereah Feb 27 at 7:31