# American Institute of Mathematical Sciences

2011, 2011(Special): 568-577. doi: 10.3934/proc.2011.2011.568

## A natural differential operator on conic spaces

 1 Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany

Received  July 2010 Revised  January 2011 Published  October 2011

We introduce the notion of a conic space, as a natural structure on a manifold with boundary, and de ne a natural fi rst order di fferential 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 de ned 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.
Citation: Daniel Grieser. A natural differential operator on conic spaces. Conference Publications, 2011, 2011 (Special) : 568-577. doi: 10.3934/proc.2011.2011.568
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