Abstract
A characteristic of a special case of Riemannean barycenters on the unit circle is presented. The non-uniqueness of such barycenters leads to an interesting study of the so-called multiple barycenters. In this work, we deal with a smooth one-dimensional manifold S1 only. Some theoretical and computational analysis is listed.
Highlights
The Riemanean geometry is the study of curved surfaces
The first thing that will be noticed is that sometimes there is more than one minimal geodesic between two mass points
We prove that the geodesic Barycenter of two mass points is a mass point contained on the shortest path
Summary
The Riemanean geometry is the study of curved surfaces. One might have a cylinder, or a sphere. One can use a cardboard paper towel roll to study a cylinder and a globe to study a sphere In such spaces, the shortest curve between any pair of mass points on such curved surface is called a minimal geodesic or geodesic. There are many minimal geodesics between the north and south poles of a sphere This leads to derive more generalized formulas, which are true in the Euclidean space. We think that for special case, for instance smooth manifolds, we could derive similar formulas to the Euclidean case, define and study interesting properties of Riemannean Barycenters. Let x, y are two points of S1 subset of Rn. If γ is a curve such that γ (a) = x , γ (b) = y and γ (t ) ∈ S1 ∀t ∈[a,b], we said that γ is a connection between x and y. =π g= ( x, m) g= (m, y) 1 g ( x, y)= for m eiπ= and x ei2π
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
