The volume of tubes in Lie groups

Mathematics > Differential Geometry [Submitted on 16 Jun 2026] Title:The volume of tubes in Lie groups View PDF HTML (experimental)Abstract:The problem of computing the volume of tubes in riemannian manifolds goes back to Weyl and Hotelling. Here we find explicit Taylor series for the volume of a tube in a Lie group equipped with a bi-invariant metric. The coefficients are smooth valuations, given by the convolution powers of the surface area valuation. We show that the tube coefficients can be naturally described as the unique valuations given by universal formulas through the formalism of differential graded Lie and Gerstenhaber algebras; in fact, they are generated by the gauge action on the Maurer--Cartan cone in the free differential graded Lie algebra on one generator. Moreover, we introduce a new convolution product on the corresponding free Gerstenhaber algebra which is compatible with the convolution of valuations and differential forms. To complete the picture, we show that a Lie group -- not necessarily connected -- admits a smooth bi-invariant valuation, beyond the Euler characteristic and the Haar measure, if and only if it admits a bi-invariant riemannian metric. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.