2018/01/22 by Bang-Yen Chen, Chen, Bang-Yen
Mathematics · #53A07 #53C40 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53A07 #msc:53C40 #msc:53C42
paper · pdf · doi:10.48550/arxiv.1801.07196
13 pages
arxiv created 2018/01/29 · arxiv updated 2018/01/31
For a submanifold M in a Euclidean space, the tangential component xT of the position vector field x of M is the most natural vector field tangent to the Euclidean submanifold, called the canonical vector field of M. In this article, first we prove that the canonical vector field of every Euclidean submanifold is always conservative. Then we initiate the study of Euclidean submanifolds with incompressible canonical vector fields. In particular, we obtain the necessary and sufficient conditions for the canonical vector field of a Euclidean submanifold to be incompressible. Further, we provide examples of Euclidean submanifolds with incompressible canonical vector field. Moreover, we classify planar curves, surfaces of revolution and hypercylinders with incompressible canonical vector fields.