Comparing Curves, Surfaces and Higher Dimensional Shapes
In this talk, I will give a generalization of the SRVF to curves in homogeneous spaces and also introduce a diffeomorphism-invariant metric on the space of vector-valued one-forms that can be used to compare shapes of immersions in $mathbb R^n$. We will see that this particular metric on the space of one-forms connects to a distinguished metric - the Ebin metric. For the space of one-forms, we obtained many interesting geometric results under this metric. The metric can be split into three parts, which gives a family of metrics for the space of shapes. I will show some examples of comparing curves and surfaces using our methods.