  We present a new construction of tubular neighborhoods in (possibly infinite
dimensional) Riemannian manifolds M, which allows us to show that if G is an
arbitrary group acting isometrically on M, then every G-invariant submanifold
with locally trivial normal bundle has a G-invariant total tubular
neighborhood. We apply this result to the Morse strata of the Yang-Mills
functional over a closed surface. The resulting neighborhoods play an important
role in calculations of gauge-equivariant cohomology for moduli spaces of flat
connections over non-orientable surfaces.
