Browsing by Subject "differential invariant"

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    Differential invariants of maximally symmetric submanifolds
    (2008-09-16) Olver, Peter J.
    Let $G$ be a Lie group acting smoothly on a manifold $M$. A closed, nonsingular submanifold $S \subset M$ is called \is{maximally symmetric} if its symmetry subgroup $\sym S \subset G$ has the maximal possible dimension, namely $\dim \sym S = \dim S$, and hence $S = \sym S \cdot z_0$ is an orbit of $\sym S$. Maximally symmetric submanifolds are characterized by the property that all their differential invariants are constant. In this paper, we explain how to directly compute the numerical values of the differential invariants of a maximally symmetric submanifold from the infinitesimal generators of its symmetry group. The equivariant method of moving frames is applied to significantly simplify the resulting formulae. The method is illustrated by examples of curves and surfaces in various classical geometries.
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    Weighted Differential Invariant Signatures and Applications to Shape Recognition
    (2016-10) Senou, Jessica
    The weighted differential invariant signature is developed to deliver more geometrical information than the signature, by combining the signature manifold with invariant measurements that capture the size of local continuous and discrete symmetries. As a consequence, the weighted signature becomes an attractive tool for the task of distinguishing between signature congruent submanifolds, which have the property that they are globally inequivalent, yet possess identical signature manifolds. Properties and relationships between such submanifolds are discussed and how these affect the weighted signature.