  Many Euclidean Einstein manifolds possess continuous symmetry groups of at
least one parameter and we consider here a classification scheme of $d$
dimensional compact manifolds based on the existence of such a one parameter
group in terms of the fixed point sets of the isometries. We discuss
applications of such a classification scheme, including the geometric
interpretation of the entropy; there are intrinsic contributions to the entropy
from the volumes of $(d-2)$ dimensional fixed point sets and contributions
related to the cohomology structure of the orbit space of the isometry. We
consider the relevance of such a decomposition of the entropy in the context of
the no boundary proposal and cosmological processes, and generalise the
discussion to compact solutions of gravity coupled to scalar and gauge fields.
