Gauge Invariance of Degenerate Riemannian Metrics

Introduction Having applications to Form recognition in mind, we want to be able to compare shapes of surfaces in R3 in a way that does not depend on parameterizations. To accomplish such so-called gauge invariance, we defined a metric on the space of parameterized surfaces that is degenerate in the direction of reparameterization. 1 What are the surfaces under consideration? The surfaces we will consider in this note are surfaces which are diffeomorphic to the unit sphere. In other words, the unit sphere will be our model surface, and the surfaces we will consider will be those that can be modeled out of it. To be mathematically precise, these are orientable genus-0 smooth compact surfaces or, equivalently, orientable 2dimensional compact simply connected submanifolds of R3 and will be called spherical surfaces in this note. How is the unit sphere represented? The good thing about the unit sphere is that only one chart suffices to cover it almost completely. We will use spherical coordinates, with polar angle θ being greater than 0 (North Pole) and less than π (South Pole) and azimuthal angleφ being greater than or equal to 0 (Greenwich prime meridian) and less than 2π (Greenwich prime meridian again); see Figure 1.