Geodesic
Euclidean submanifolds with incompressible canonical vector field
Serdica Mathematical Journal, Tome 43 (2017) no. 3-4, pp. 321-334.

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For a submanifold M in a Euclidean space E^m, the tangential component x^T 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.
Keywords: Euclidean submanifold, canonical vector field, conservative vector field, incompressible vector field, 53A07, 53C40, 53C42 
@article{SMJ2_2017_43_3-4_a4,
     author = {Chen, Bang-Yen},
     title = {Euclidean submanifolds with incompressible canonical vector field},
     journal = {Serdica Mathematical Journal},
     pages = {321--334},
     publisher = {mathdoc},
     volume = {43},
     number = {3-4},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SMJ2_2017_43_3-4_a4/}
}
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Chen, Bang-Yen. Euclidean submanifolds with incompressible canonical vector field. Serdica Mathematical Journal, Tome 43 (2017) no. 3-4, pp. 321-334. http://geodesic.mathdoc.fr/item/SMJ2_2017_43_3-4_a4/