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Journal of Lie Theory 26 (2016), No. 1, 235--267
Copyright Heldermann Verlag 2016

The Symmetry Groupoid and Weighted Signature of a Geometric Object

Peter J. Olver
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A.

We refine the concept of the symmetry group of a geometric object through its symmetry groupoid, which incorporates both global and local symmetries in a common framework. The symmetry groupoid is related to the weighted differential invariant signature of a submanifold, that is introduced to capture its fine grain equivalence and symmetry properties. Applications to the recognition and symmetry properties of digital images are indicated.

Keywords: Coarea formula, differential invariant, equivalence, global symmetry, groupoid, index, local symmetry, moving frame, piece, signature, submanifold, syzygy, weighted signature.

MSC: 18B40, 20L05, 22A22, 53A04, 53A05, 53A55

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