  We generalize the notion of Gaussian bridges by conditioning Gaussian
processes given that certain linear functionals of the sample paths vanish. We
show the equivalence of the laws of the unconditioned and the conditioned
process and by an application of Girsanov's theorem, we show that the
conditioned process follows a stochastic differential equation (SDE) whenever
the unconditioned process does. In the Markovian case, we are able to determine
the coefficients in the SDE of the conditioned process explicitly. Our main
example is Brownian motion on $[0,1]$ pinned down in 0 at time 1 and
conditioned to have vanishing area spanned by the sample paths.
