  Working in the Palatini formalism, we describe a procedure for constructing
degenerate solutions of general relativity on 4-manifold M from certain
solutions of 2-dimensional BF theory on any framed surface Sigma embedded in M.
In these solutions the cotetrad field e (and thus the metric) vanishes outside
a neighborhood of Sigma, while inside this neighborhood the connection A and
the field E = e ^ e satisfy the equations of 4-dimensional BF theory. Moreover,
there is a correspondence between these solutions and certain solutions of
2-dimensional BF theory on Sigma. Our construction works in any signature and
with any value of the cosmological constant. If M = R x S for some 3-manifold
S, at fixed time our solutions typically describe `flux tubes of area': the
3-metric vanishes outside a collection of thickened links embedded in S, while
inside these thickened links it is nondegenerate only in the two transverse
directions. We comment on the quantization of the theory of solutions of this
form and its relation to the loop representation of quantum gravity.
