ISOMETRIC IMMERSIONS IN 3-DIMENSIONAL EUCLIDEAN SPACE

Year 2021, Volume: 9 Issue: 1, 20 - 24, 22.02.2021

Emre Öztürk

https://doi.org/10.20290/estubtdb.648030

Abstract

In this paper, we examine the image of geodesic curves of Riemann 2-manifolds under the isometric immersions, in three dimensional Euclidean space. We show that the curvature of these curves is equal to the normal curvature of the manifold in the direction of tangent vector field of the geodesics. Moreover, we prove that if the parameter curves of the manifold are the line of curvature, then the geodesic torsion of geodesics is equal to the torsion of the image curve.

Keywords

Immersion,, Geodesic,, Riemann manifold

ISOMETRIC IMMERSIONS IN 3-DIMENSIONAL EUCLIDEAN SPACE

Abstract

In
this paper, we examine the image of geodesic curves of Riemann 2-manifolds
under the isometric immersions, in three dimensional Euclidean space. We show
that the curvature of these curves is equal to the normal curvature of the manifold in the direction of
tangent vector field of the geodesics. Moreover, we prove that if the parameter
curves of the manifold are the line of curvature, then the geodesic torsion of geodesics
is equal to the torsion of the image curve. 

Keywords

Immersion, Geodesic, Riemann manifold

References

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• O’Neill B. Elementary Differential Geometry. 2nd ed. New York: Academic Press, 2006.
• Sabuncuoglu A. Diferensiyel Geometri. Ankara: Nobel Yayın Dağıtım, 2006.