The geometry of a Randers rotational surface

Hama, Rattanasak; Chitsakul, Pakkinee; Sabau, Sorin V.

doi:10.5486/pmd.2015.7395

Cited by 6 publications

(15 citation statements)

References 13 publications

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“…Moreover, we show here that the flag curvature of this Randers metric coincide with the Gaussian curvature of h (Lemma 2.12). Even though some of these results were proved already in [7], for a surface of revolution homeomorphic to R 2 , we show here how they extend to a 2-sphere of revolution.…”

Section: A Single Point On the Antipodal Parallelmentioning

confidence: 56%

“…Proof. Even though similar with the proof of Lemma 4.3 in [7] we sketch it here for the sake of completeness. We can see that our Randers rotational surface of revolution is Finsler-Einstein with Ricci scalar Ric (F ) = K(x), where K(x) is the sectional curvature of (M, F ).…”

Section: Randers Rotational Metricsmentioning

confidence: 97%

“…In a previous paper [7] we have constructed a Randers rotational metric on a surface of revolution homeomorphic to R 2 . We will construct a Randers rotational metric on a 2-sphere of revolution in a similar manner in the following.…”

Section: Randers Rotational Metricsmentioning

confidence: 99%

“…where ψ(s) is the angle between the vectorsṖ(s) and ∂ ∂θ | P(s) . We have proved this formula for a surface of revolution homeomorphic to R 2 in [7], but the proof carries out on any kind of surface of revolution. 1.…”

Section: Randers Rotational Metricsmentioning

confidence: 99%

“…We recall that the wind is blowing along the parallels (see [7]). Since d h ( p 0 , q 0 ) = π, we know that the F -unit speed geodesic P + (s) = ϕ(s, γ(s)) obtained by γ(s) is joining p 0 to the point P + (π) = ϕ(π, γ(π)) = (a, π(1 + µ)), therefore the point q 0 will change the position to q 1 = (a, π(1 + µ)), hence d F ( p 0 , q 1 ) = π.…”

Section: Randers Rotational Metricsmentioning

confidence: 99%

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The cut locus of a Randers rotational 2-sphere of revolution

Hama¹,

Kasemsuwan²,

Sabau³

2018

Publ. Math. Debrecen

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In the present paper we study structure of the cut locus of a Randers rotational 2-sphere of revolution (M, F = α + β). We show that in the case when Gaussian curvature of the Randers surface is monotone along a meridian the cut locus of a point q ∈ M is a point on a subarc of the opposite half bending meridian or of the antipodal parallel (Theorem 1.1). More generally, when the Gaussian curvature is not monotone along the meridian, but the cut locus of a point q on the equator is a subarc of the same equator, then the cut locus of any point q ∈ M different from poles is a subarc of the antipodal parallel (Theorem 1.2). Some examples are also given at the last section. * Mathematics Subject Classification (2010) : 53C60, 53C22. † Keywords: Randers metrics, 2-sphere of revolution, cut locus, Gaussian curvature.2 The 2-sphere of revolution 2.1 The Riemannian 2-sphere of revolution A compact Riemannian manifold (M, h) homeomorphic to a 2-sphere is called a 2sphere of revolution if M admits a point p, called pole, such that for any two points q 1 , q 2 on M with d h (p, q 1 ) = d h (p, q 2 ), there exists an h-isometry f on M satisfying f (q 1 ) = q 2 , and f (p) = p, where d h (·, ·) denoted the h-Riemannian distance function on M.Let (r, θ) denote geodesic polar coordinates around a pole p of (M, h). The Riemannian metric can be expressed as h = dr 2 + m 2 (r)dθ 2 on M \ {p, q}, where q denotes the unique

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Section: A Single Point On the Antipodal Parallelmentioning

confidence: 56%

Section: Randers Rotational Metricsmentioning

confidence: 97%

Section: Randers Rotational Metricsmentioning

confidence: 99%

Section: Randers Rotational Metricsmentioning

confidence: 99%

Section: Randers Rotational Metricsmentioning

confidence: 99%

See 3 more Smart Citations