Rotation Minimizing Vector Fields and Frames in Riemannian Manifolds

Etayo, Fernando

doi:10.1007/978-3-319-32085-4_8

Cited by 12 publications

(19 citation statements)

References 14 publications

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“…However, we can also consider any other adapted orthonormal moving frame along α(s): the equation of motion of such a moving frame is then given by a skew-symmetric matrix. Of particular importance are the so-called Rotation Minimizing (RM) Frames [2,14]: we say that {t, n 1 , . .…”

Section: Preliminariesmentioning

confidence: 99%

“…. , κ m uniquely determines a curve up to rigid motions of E m+1 [2,14]. In addition, a remarkable advantage of using RM frames is that they allow for a simple characterization of spherical and plane curves:…”

Section: Preliminariesmentioning

confidence: 99%

“…It is also possible to introduce Frenet frames in Riemannian manifolds [19,30], see also [4,19,23,24,31]. Analogously, one can also define RM frames [14,15]. To introduce such concepts, one should take covariant derivatives in the direction of the unit tangent instead of the ordinary one.…”

Section: Rotation Minimizing Frames and Normal Curves In Riemannian Gmentioning

confidence: 99%

“…, n m } along α : I → R m+1 is characterized by the equations t ′ (s) = m i=1 κ i (s)n i (s) and n ′ i (s) = −κ i (s)t(s), where s is an arc-length parameter. The basic idea here is that n i rotates only the necessary amount to remain normal to t: in fact, n i is parallel transported along α with respect to the normal connection [14]. Due to their minimal twist, RM frames are of importance in applications, such as in computer graphics and visualization [16,33], sweep surface modeling [3,27,29], and in differential geometry as well [2,11,12,15], just to name a few.…”

Section: Introductionmentioning

confidence: 99%

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Characterization of Curves that Lie on a Geodesic Sphere or on a Totally Geodesic Hypersurface in a Hyperbolic Space or in a Sphere

Silva

2018

Mediterr. J. Math.

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The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion. In this work, we extend these investigations to characterize curves that lie on a geodesic sphere or totally geodesic hypersurface in a Riemannian manifold of constant curvature. Using that geodesic spherical curves are normal curves, i.e., they are the image of an Euclidean spherical curve under the exponential map, we are able to characterize geodesic spherical curves in hyperbolic spaces and spheres through a non-homogeneous linear equation. Finally, we also show that curves on totally geodesic hypersurfaces, which play the role of hyperplanes in Riemannian geometry, should be characterized by a homogeneous linear equation. In short, our results give interesting and significant similarities between hyperbolic, spherical, and Euclidean geometries.Mathematics Subject Classification (2010) 53A04 · 53A05 · 53B20 · 53C21

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Rotation Minimizing Frames and Normal Curves In Riemannian Gmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Characterization of Curves that Lie on a Geodesic Sphere or on a Totally Geodesic Hypersurface in a Hyperbolic Space or in a Sphere

Silva

2018

Mediterr. J. Math.

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“…The above condicion is equivalent to the fact ∇ α ′ N and α ′ are proportional (see [8] for the details). As the normal connection is also metric, one can conclude that the norm of an RM vector field is constant and that the angle between two RM vector fields remains constant.…”

Section: Introductionmentioning

confidence: 99%

Geometric properties of rotation minimizing vector fields along curves in Riemannian manifolds

Etayo¹

2018

Turk J Math

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Rotation minimizing (RM) vector fields and frames were introduced by Bishop as an alternative to the Frenet frame. They are used in CAGD because they can be defined even when the curvature vanishes. Nevertheless, many other geometric properties have not been studied. In the present paper, RM vector fields along a curve immersed into a Riemannian manifold are studied when the ambient manifold is the Euclidean 3-space, the hyperbolic 3-space, and a Kähler manifold.

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On calculation of the pitch and angle of pitch of a closed ruled surface of dimension (k + 1) using a rotation minimizing frame in Euclidean space ℝn

Keskin

Altın

Ekmekci

et al. 2021

Math Methods in App Sciences

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In this paper, first, the pitch and the angle of pitch of a closed ruled surface of dimension (k + 1) are calculated according to a rotation minimizing frame in Euclidean space R 4 . To calculate this pitch and this angle of pitch of the closed ruled surface of dimension (k + 1), three different ruled surfaces are taken according to the rotation minimizing frame. Therefore, for these three different ruled surfaces, three different situations are found. Second, by taking any curve in four-dimensional Euclidean space, the closed ruled surface of this curve, its orthogonal trajectory, the pitch, and the angle of pitch of this closed ruled surface are obtained according to the rotation minimizing frame. Finally, the pitch and the angle of pitch of the closed ruled surface of dimension (k + 1) are generalized to n-dimensional Euclidean space with the help of the rotation minimizing frame.

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Rotation Minimizing Vector Fields and Frames in Riemannian Manifolds

Abstract: We prove that a normal vector field along a curve in R 3 is rotation minimizing (RM) if and only if it is parallel respect to the normal connection. This allows us to generalize all the results of RM vectors and frames to curves immersed in Riemannian manifolds.

Cited by 12 publications

References 14 publications

Characterization of Curves that Lie on a Geodesic Sphere or on a Totally Geodesic Hypersurface in a Hyperbolic Space or in a Sphere

Characterization of Curves that Lie on a Geodesic Sphere or on a Totally Geodesic Hypersurface in a Hyperbolic Space or in a Sphere

Geometric properties of rotation minimizing vector fields along curves in Riemannian manifolds

On calculation of the pitch and angle of pitch of a closed ruled surface of dimension (k + 1) using a rotation minimizing frame in Euclidean space ℝn

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