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### Open Access Journals

PROC

We introduce the notion of a conic space, as a natural structure on
a manifold with boundary, and dene a natural first order differential operator,
$c_d_\partial$, acting on boundary values of conic one-forms. Conic structures arise, for
example, from resolutions of manifolds with conic singularities, embedded in a
smooth ambient space. We show that pull-backs of smooth ambient one-forms
to the resolution are cd@-closed, and that this is the only local condition on oneforms
that is invariantly dened on conic spaces. The operator $c_d_\partial$ extends
to conic Riemannian metrics, and $c_d_\partial$-closed conic metrics have important
geometric properties like the existence of an exponential map at the boundary.

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