Conformal image of an osculating curve on a smooth immersed surface
aa r X i v : . [ m a t h . G M ] J u l CONFORMAL IMAGE OF AN OSCULATING CURVE ON ASMOOTH IMMERSED SURFACE
ABSOS ALI SHAIKH , MOHAMD SALEEM LONE AND PINAKI RANJAN GHOSH Abstract.
The main intention of the paper is to investigate an osculatingcurve under the conformal map. We obtain a sufficient condition for the con-formal invariance of an osculating curve. We also find an equivalent systemof a geodesic curve under the conformal transformation(motion) and show itsinvariance under isometry and homothetic motion. Introduction
The study of smooth maps between surfaces is an important field of study indifferential geometry. There are various mappings which preserve certain differ-ential geometric quantities. Depending on the invariance of the mean curvatureand the Gaussian curvature, we mainly classify the transformations as isometric,conformal and non-conformal. In case of isometry both length as well as the angleis preserved. A geometrical way of interpreting isometry is that it preserves theGaussian curvature but not the mean curvature. One of the examples of such atransformation is the existence of an isometry between a helicoid and a catenoid. Ageneralized class of isometry is called a conformal transformation which is a dilatedform of isometry. In the case of conformal motion, only angles are preserved andnot necessarily distances. An important example of conformal transformation is astereographic projection. Gerardus Mercator used the conformal property of stere-ographic projection to develop the famous Mercator’s world map. It is believedthe world’s first angle preserving map. A beautiful explanation together with someapplications of conformal maps, Bobenko and Gunn recently published a moviewith Springer on conformal maps, we strongly appeal the reader to see [1]. A moregeneralized class of motions is of non-conformal transformations, wherein neitherdistances nor angles are preserved.The geometric position of an arbitrary point on a curve or on a surface is affirmedby the position vector field. In case of curve, the position vector field can be thoughtof a trajectory of the curve, wherein its first derivative gives the velocity and thesecond derivative gives the acceleration of the trajectory. In this paper, we aregoing to study the conformal properties of a curve which totally depends uponits position vector field. If the position vector field of a curve always lies in theorthogonal complement of the binormal vector, we say that type of curve as anosculating curve. In other words an osculating curve is a curve whose positionvector always lies in the osculating plane spanned by unit tangent vector and unitnormal vector. Similarly we can define normal curve as a curve whose position
Mathematics Subject Classification.
Key words and phrases.
Osculating curve, conformal map, homothetic map, normal curvature,geodesic curvature. , MOHAMD SALEEM LONE AND PINAKI RANJAN GHOSH vector lies in the plane spanned by the unit normal and binormal vector. Chen([2]) introduced rectifying curves and obtained a relation between a rectifying curveand the ratio of its curvature and torsion. Chen ([2]) characterized the rectifyingcurve as the ratio of its curvature and torsion is a non-constant linear function ofthe parameter. For details we refer the reder to see [3, 4].The motivation of this paper starts with the article [8]. In [8] Shaikh and Ghoshinvestigated the invariant properties of a rectifying curve lying on a smooth im-mersed surface under isometric transformation. In that paper, they obtained asufficient condition with respect to which a rectifying curve retains its rectifyingproperty under an isometric motion. In addition to this sufficient condition, theyproved that the normal component of the rectifying curve is preserved under thesame isometric motion. This paper laid a foundation for many problems, for exam-ple, what happens to an osculating or a normal curve lying on smooth immersedsurfaces under the isometric motion? Both of these problems were discussed in[7, 9, 10]. Later in [11], we generalize the notion of study by conformal transforma-tion, wherein we study the conformal transformation of rectifying curves lying onsmooth surfaces. In this paper([11]), in addition to various geometric invariants,we obtain a sufficient condition with respect to which a rectifying curve retains itsnature under conformal motion. Now, the first hand possible studies dependingupon the position vector field and the conformality will be the investigation of os-culating and normal curves. The present paper is devoted to investigate osculatingcurves and the question with respect to the normal curve can be a future problem.The paper is framed as follows. Section 2 is devoted to some rudimentary factsabout the curves lying on a smooth immersed surface. In section 3, we discussthe main results. We obtain a sufficient condition for the conformal invarianceof an osculating curve. We also find a condition for the homothetic invariance ofthe normal component of the osculating curve and prove that the component of anosculating curve along any tangent vector to the surface is homothetic invariant. Wealso find an equivalent system of a geodesic curve under conformal transformationand show its invariance under isometry and homothetic motion.2. Preliminaries
Suppose ~t , ~n and ~b are respectively the tangent, normal and binormal vectorsat any point to a unit speed curve σ : I ⊂ R → E such that { ~t, ~n, ~b } forms aSerret-Frenet frame. If ′ denotes the differentiation with respect to the arc lengthparameter s , then we have d~tds = κ~n d~nds = − κ~t + τ~b d~bds = − τ~n, where κ and τ are respectively the curvature and torsion of the curve σ . Definition 2.1.
A diffeomorphism G : M → ˜ M is said to be local isometry betweenthe surfaces M and ˜ M , if for all x , x ∈ T p ( M ) and for any p ∈ M , h x , x i p = h d G p ( x ) , d G p ( x ) i G ( p ) . A diffeomorphism which in addition to local isometry is a bijective map is calledan isometry. If there exists such an isometry G : M → ˜ M , then M and ˜ M are saidto be isometric. ONFORMAL IMAGE OF AN OSCULATING CURVE ON A SMOOTH IMMERSED SURFACE3 If G : M → ˜ M is a local isometry, then the coefficients of their first fundamentalforms are invariant under G and hence, E = ˜ E, F = ˜
F , G = ˜ G. Definition 2.2.
Let G be a diffeomorphism between two smooth surfaces M and˜ M . Then G is said to be a local conformal map between M and ˜ M , if for all x , x ∈ T p ( M ) with an arbitrary p ∈ M , we have δ h d G p ( x ) , d G p ( x ) i G ( p ) = h x , x i p , where δ is a differentiable function on M , also known as dilation factor. We seethat a conformal motion is the composition of an isometry and a dilation. Weobserve that if the dilation factor is identity, then we get the isometry. Geomet-rically, we can say that the conformal maps preserve angles both in direction andmagnitude but not necessarily the lengths. In this case [5]: δ E = ˜ E, δ F = ˜ F , δ G = ˜ G. Here we shall call that the first fundamental form coefficients are conformally in-variant.For such a G to be conformal, a necessary and sufficient condition is that theirline elements are proportional and the ratio of arc elements given by dsd ˜ s is equal tothe dilation factor δ . If for all points on the surface the dilation factor is a non-zeroconstant(say c ), then the conformal map is said to be homothetic. If the dilationfunction is constantly equal to one, the conformal map becomes an isometry. Thuswe can say isometric maps are a subset of conformal maps with the dilation factor δ = 1 [6]. Definition 2.3.
Let M and ˜ M be two smooth surfaces with Φ( u, v ) being thesurface patch of M and let g : M → ˜ M be a smooth map such that ˜ g = g ◦ Φ: • If g is conformal, then g is said to be conformally invariant when ˜ g = δ g for some dilation factor δ ( u, v ) . • If g is homothetic, then g is said to be homothetic invariant when ˜ g = c g, ( c = { , } ). Definition 2.4.
Let σ : I ⊂ R → E be a smooth curve. Then σ is said to bean osculating curve if its position vector is lying in the orthogonal complement ofnormal vector i.e., σ · ~n = 0 , and hence(2.1) σ ( s ) = ξ ( s ) ~t ( s ) + µ ( s ) ~n ( s ) , where ξ, µ are two smooth functions.Let M be a smooth surface with Φ( u, v ) : U ⊂ R → M being its chart map([page no 52, [5]]). Then, for the curve σ ( s ) = Φ( u ( s ) , v ( s )) on M , we have σ ′ ( s ) = Φ u u ′ + Φ v v ′ , or, ~t ( s ) = σ ′ ( s ) = Φ u u ′ + Φ v v ′ (2.2) ~t ′ ( s ) = u ′′ Φ u + v ′′ Φ v + u ′ Φ uu + 2 u ′ v ′ Φ uv + v ′ Φ vv . ABSOS ALI SHAIKH , MOHAMD SALEEM LONE AND PINAKI RANJAN GHOSH If N is the surface normal, then(2.3) ~n ( s ) = 1 κ ( s ) ~t ′ ( s ) = 1 k ( s ) (Φ u u ′′ + Φ v v ′′ + Φ uu u ′ + Φ uv u ′ v ′ + Φ vv v ′ ) .~b ( s ) = 1 k ( s ) h (Φ u u ′ + Φ v v ′ ) × (Φ u u ′′ + Φ v v ′′ + Φ uu u ′ + Φ uv u ′ v ′ + Φ vv v ′ ) i , = 1 k ( s ) h { u ′ v ′′ − u ′′ v ′ } N + Φ u × Φ uu u ′ + 2Φ u × Φ uv u ′ v ′ + Φ u × Φ vv u ′ v ′ +Φ v × Φ uu u ′ v ′ + 2Φ v × Φ uv u ′ v ′ + Φ v × Φ vv v ′ i . Definition 2.5. If σ be a unit speed curve on M , then ~t ⊥ N and hence N × σ ′ , σ ′ and N are mutually orthogonal to each other such that σ ′′ = κ g N × σ ′ + κ n N ,where κ g and κ n are respectively known as the geodesic and normal curvature of σ . Again since σ ′′ = κ ( s ) ~n ( s ), we have κ n = κ ( s ) ~n ( s ) · N = ( u ′′ Φ u + v ′′ Φ v + u ′ Φ uu + 2 u ′ v ′ Φ uv + v ′ Φ vv ) · N or(2.4) κ n = u ′ L + 2 u ′ v ′ M + v ′ N, where L, M, N are the coefficients of the second fundamental form of the surface.We say that σ is asymptotic iff κ n = 0 . Conformal image of an osculating curve.
Let σ ( s ) be an osculating curve lying on a smooth immersed surface M in E ,then with the help of (2.1), (2.2) and (2.3), we get(3.1) σ ( s ) = ξ ( s )(Φ u u ′ +Φ v v ′ )+ µ ( s ) k ( s ) h u ′′ Φ u + v ′′ Φ v + u ′ Φ uu +2 u ′ v ′ Φ uv + v ′ Φ vv i . Theorem 3.1. If G : M → ˜ M is a conformal map, then the image ˜ σ ( s ) of σ ( s ) under G is an osculating curve on ˜ M when ˜ σ − δ G ∗ ( σ ) = µκ h u ′ (cid:18) δ u G ∗ Φ u + δ ∂ G ∗ ∂u Φ u (cid:19) + 2 u ′ v ′ (cid:18) δ u G ∗ Φ v + δ ∂ G ∗ ∂u Φ v (cid:19) + v ′ (cid:18) δ v G ∗ Φ v + δ ∂ G ∗ ∂v Φ v (cid:19) i . (3.2) Proof.
Let ˜ M be the conformal image of M and Φ( u, v ) and ˜Φ( u, v ) = G ◦ Φ( u, v ) bethe surface patches of M and ˜ M , respectively. Then the differential map d G = G ∗ of G sends each vector of the tangent space T p M to a dilated tangent vector of thetangent space of T G ( p ) ˜ M with the dilation factor δ .˜Φ u ( u, v ) = δ ( u, v ) G ∗ Φ u , (3.3) ˜Φ v ( u, v ) = δ ( u, v ) G ∗ Φ v . (3.4) ONFORMAL IMAGE OF AN OSCULATING CURVE ON A SMOOTH IMMERSED SURFACE5
Differentiating (2 .
2) and (2 .
3) partially with respect to both u and v respectively,we get ˜Φ uu = δ u G ∗ Φ u + δ ∂ G ∗ ∂u Φ u + δ G ∗ Φ uu ˜Φ vv = δ v G ∗ Φ v + δ ∂ G ∗ ∂v Φ v + δ G ∗ Φ vv (3.5) ˜Φ uv = δ u G ∗ Φ v + δ ∂ G ∗ ∂u Φ v + δ G ∗ Φ uv = δ v G ∗ Φ u + δ ∂ G ∗ ∂v Φ u + δ G ∗ Φ uv . Therefore in view of (3.2), (3.3), (3.4) and (3.5), we obtain˜ σ = ξ ( u ′ δ G ∗ Φ u + v ′ δ G ∗ Φ v ) + µκ h u ′ (cid:18) δ u G ∗ Φ u + δ ∂ G ∗ ∂u Φ u + δ G ∗ Φ uu (cid:19) +2 u ′ v ′ (cid:18) δ u G ∗ Φ v + δ ∂ G ∗ ∂u Φ v + δ G ∗ Φ uv (cid:19) + v ′ (cid:18) δ v G ∗ Φ v + δ ∂ G ∗ ∂v Φ v + δ G ∗ Φ vv (cid:19) i , which can be written as˜ σ ( s ) = ξ ( s )( ˜Φ u u ′ + ˜Φ v v ′ ) + µ ( s ) k ( s ) h u ′′ ˜Φ u + v ′′ ˜Φ v + u ′ ˜Φ uu + 2 u ′ v ′ ˜Φ uv + v ′ ˜Φ vv i , i.e., ˜ σ ( s ) = ˜ ξ ( s )˜ ~t ( s ) + ˜ µ ( s )˜ κ ( s ) ˜ ~n ( s )for some C ∞ functions ˜ ξ ( s ) and ˜ µ ( s ) . Thus ˜ σ ( s ) is an osculating curve. (cid:3) Here and now on onward, we accept ˜ ξ , ˜ µ and ˜ κ are also dilated with δ in such away, such that ˜ µ/ ˜ κ = µ/κ. Corollary 3.2.
Let M and ˜ M be two smooth surfaces and G be a homothetic mapbetween them and σ ( s ) be an osculating curve on M . Then ˜ σ ( s ) is an osculatingcurve on ˜ M if ˜ σ − c G ∗ ( σ ) = µκ h u ′ (cid:18) c ∂ G ∗ ∂u Φ u (cid:19) + 2 u ′ v ′ (cid:18) c ∂ G ∗ ∂u Φ v (cid:19) + v ′ (cid:18) c ∂ G ∗ ∂v Φ v (cid:19) i . Corollary 3.3. [8]
Let G : M → ˜ M be an isometry between two smooth immersedsurfaces M and ˜ M in E and σ ( s ) be an osculating curve on M . Then ˜ σ ( s ) is anosculating curve on ˜ M if ˜ σ − G ∗ ( σ ) = µκ h u ′ ∂ G ∗ ∂u Φ u + 2 u ′ v ′ ∂ G ∗ ∂u Φ v + v ′ ∂ G ∗ ∂v Φ v i . Theorem 3.4.
If the smooth surfaces M and ˜ M are conformally related and σ ( s ) is a non-asymptotic osculating curve on M , then for the component of σ along thesurface normal, the following relation holds: ˜ σ · ˜ N − δ ( σ · N ) = µκ δ W h u ′ ( ˜Φ uu − δ Φ uu )+2 u ′ v ′ ( ˜Φ uv − δ Φ uv )+ v ′ ( ˜Φ vv − δ Φ vv ) i . ABSOS ALI SHAIKH , MOHAMD SALEEM LONE AND PINAKI RANJAN GHOSH Proof.
Let σ ( s ) be an osculating curve on M whose at least second order partialderivatives are non-vanishing. To find the position vector of σ along the normal N to the surface M at a point σ ( s ), from (3.1), we have(3.6) σ ( s ) · N = µ ( s ) k ( s ) h u ′ L + 2 u ′ v ′ M + v ′ N i , where L, M and N are the second fundamental form coefficients. In the Mongepatch form, the coefficients are given by(3.7) L = Φ uu u + Φ v (= W ) , M = Φ uv u + Φ v , N = Φ vv u + Φ v . Now if ˜ σ ( a ) is an osculating curve on ˜ M , we have(3.8) ˜ σ · ˜ N = µ ( s ) k ( s ) h u ′ ˜ L + 2 u ′ v ′ ˜ M + v ′ ˜ N i . Again by the Monge patch form, the conformal coefficients are given by(3.9) ˜ L = ˜Φ uu u + ˜Φ v , ˜ M = ˜Φ uv u + ˜Φ v , ˜ N = ˜Φ vv u + ˜Φ v . From (3.6) and (3.8), we get˜ σ · ˜ N − δ ( σ · N ) = µκ h u ′ ( ˜ L − δ L ) + 2 u ′ v ′ ( ˜ M − δ M ) + v ′ ( ˜ N − δ N ) i . Using (3.9) the above equation turns out to be(3.10)˜ σ · ˜ N − δ ( σ · N ) = µκ δ W h u ′ ( ˜Φ uu − δ Φ uu )+2 u ′ v ′ ( ˜Φ uv − δ Φ uv )+ v ′ ( ˜Φ vv − δ Φ vv ) i , which proves the theorem. (cid:3) Corollary 3.5. If M and ˜ M are conformally related such that σ ( s ) is osculatingcurve on M , then the normal component of σ ( s ) is invariant under the conformalmap if any one of the following relation holds: (i) The position vector of σ ( s ) is in the direction of the tangent vector. (ii) The curve σ ( s ) is asymptotic. (iii) The normal curvature is invariant under the confomal map.Proof.
From (2.4), (3.6) and (3.8), we get(3.11) ˜ σ · ˜ N − δ ( σ · N ) = µ (˜ κ n − δ κ n ) κ . Now ˜ σ · N − δ ( σ · N ) = 0 if and only if µ = 0 and ˜ κ n − δ κ n = 0 , implying σ ( s ) = ξ ( s ) t ( s ) which proves ( i ) of the corollary. ( ii ) and ( iii ) are the directimplications of ˜ κ n − δ κ n = 0. (cid:3) Corollary 3.6. If M and ˜ M are homothetic and σ ( s ) is an osculating curve on M , then the normal component of σ ( s ) is homothetically invariant if any one ofthe following relation holds: (i) ( i ) and ( ii ) of corollary . holds. (ii) The normal curvature is homothetic invariant.
Corollary 3.7. If M and ˜ M are isometric and σ ( s ) is an osculating curve on M ,then the component of σ along the surface normal is invariant under isometry ifany one of the following relation holds: ONFORMAL IMAGE OF AN OSCULATING CURVE ON A SMOOTH IMMERSED SURFACE7 (i) ( i ) and ( ii ) of corollary . holds. (ii) The normal curvature is invariant under isometry.
Theorem 3.8. If G : M → ˜ M is a conformal map and σ ( s ) is an osculating curveon M , then for the tangential component we have ˜ σ ( s ) · T = δ ( σ · T ) + h ( E, F, G, δ ) , (3.12) where h ( E, F, G, δ ) = µ κ h a (cid:16) u ′ δδ u E + 4 u ′ v ′ δδ v E + 4 v ′ δδ v F − v ′ δδ u G (cid:17) + b (cid:16) u ′ δδ u F − u ′ δδ v E + 4 v ′ u ′ δδ u G + 2 v ′ δδ v G (cid:17) i . (3.13) Proof.
Let ˜ M be the conformal image of M and Φ( u, v ) and ˜Φ( u, v ) = G ◦ Φ( u, v )be the surface patches of M and ˜ M , respectively. We know that(3.14) δ E = ˜ E, δ F = ˜ F , δ G = ˜ G. This implies that ˜ E u = 2 δδ u E + δ E u , ˜ E v = 2 δδ v E + δ E v , ˜ F u = 2 δδ u F + δ F u , ˜ F v = 2 δδ v F + δ F v , ˜ G u = 2 δδ u G + δ G u , ˜ G v = 2 δδ v G + δ G v . (3.15)Now, we have E u = (Φ u · Φ u ) u = 2Φ uu · Φ u (3.16) Φ uu · Φ u = E u . Similarly, it is easy to check thatΦ uu · Φ v = F u − E v , Φ uv · Φ u = E v , Φ uv · Φ v = G u , Φ vv · Φ v = G v , Φ vv · Φ u = F v − G u . (3.17)Thus from (3.1), (3.16) and (3.17), we can easily deduce(3.18) σ ( s ) · Φ u = ξ ( s )( u ′ E + v ′ F )+ µ ( s )2 κ ( s ) [2 u ′′ E +2 v ′′ F + u ′ E u +2 u ′ v ′ E v +2 v ′ F v − v ′ G u ] . Similarly(3.19) σ ( s ) · Φ v = ξ ( s )( u ′ F + v ′ G )+ µ ( s )2 κ ( s ) [2 u ′′ F +2 v ′′ G +2 u ′ F u − u ′ E v +2 v ′ u ′ G u + v ′ G v ] . Now if ˜ σ be an osculating curve on ˜ M and T = a Φ u + b Φ v be the tangent vectorof ˜ M at ˜ σ ( s ), we have˜ σ ( s ) · ( a ˜Φ u + b ˜Φ v ) = a h ˜ ξ ( s )( u ′ ˜ E + v ′ ˜ F ) + ˜ µ ( s )2˜ κ ( s ) (cid:16) u ′′ ˜ E + 2 v ′′ ˜ F + u ′ ˜ E u +2 u ′ v ′ ˜ E v + 2 v ′ ˜ F v − v ′ ˜ G u (cid:17)i + b h ˜ ξ ( s )( u ′ ˜ F + v ′ ˜ G )+ ˜ µ ( s )2˜ κ ( s ) (cid:16) u ′′ ˜ F + 2 v ′′ ˜ G + 2 u ′ ˜ F u − u ′ ˜ E v + 2 v ′ u ′ ˜ G u + v ′ ˜ G v (cid:17)i , ABSOS ALI SHAIKH , MOHAMD SALEEM LONE AND PINAKI RANJAN GHOSH or˜ σ ( s ) · T − δ [ σ ( s ) · T ] = µ κ h a (cid:16) u ′ δδ u E + 4 u ′ v ′ δδ v E + 4 v ′ δδ v F − v ′ δδ u G (cid:17) + b (cid:16) u ′ δδ u F − u ′ δδ v E + 4 v ′ u ′ δδ u G + 2 v ′ δδ v G (cid:17) i . (cid:3) Corollary 3.9.
Let G : M → ˜ M be a homothetic conformal map between twosmooth surfaces and σ ( s ) be an osculating curve on M . Then the tangential com-ponent is homothetic invariant.Proof. The claim directly follows from (3.12) and (3.13) while assuming δ ( u, v ) = c . (cid:3) Corollary 3.10. [8]
Let G : M → ˜ M be an isometry and σ ( s ) be an osculatingcurve on M . Then the tangential component of σ remains invariant. Theorem 3.11.
Let G : M → ˜ M be a conformal map between two smooth surfacesand σ ( s ) be an osculating curve on M . Then for the geodesic curvature, we have (3.20) ˜ κ g = δ κ g + f ( E, F, G, δ ) , where f ( E, F, G, δ ) = h ǫ u ′ + (2 ǫ − ǫ ) u ′ v ′ + ( ǫ − ǫ ) u ′ v ′ − ǫ v ′ ip EG − F . Proof.
Let σ ( s ) be an osculating curve on M and ˜ σ = G ◦ σ be the conformal imageof σ on ˜ M . From the definition of geodesic curvature κ g = σ ′′ · ( N × σ ′ ) . In [7] Shaikh and Ghosh showed that for such a σ , we have κ g = u ′ v ′′ ( EF − F E ) + v ′ u ′′ ( F − GE ) + u ′ v ′′ ( EG − F ) + v ′ v ′′ ( F G − GF )+ u ′ ( E Φ uu · Φ v − F Φ uu · Φ u ) + u ′ v ′ ( F Φ uu · Φ v − G Φ uu · Φ u )+2 u ′ v ′ ( E Φ uv · Φ v − F Φ uv · Φ u ) + 2 u ′ v ′ ( F Φ uv · Φ v − G Φ uv · Φ u )+ u ′ v ′ ( E Φ vv · Φ v − F Φ vv · Φ u ) + v ′ ( F Φ vv · Φ v − G Φ vv · Φ u ) . With the help of (3.16) and (3.17), the above equation turns out to be κ g = ( u ′ v ′′ − v ′ u ′′ )( EG − F ) + 12 u ′ (2 EF u − EE v − F E u )+ 12 u ′ v ′ (2 F F u − F E v − GE u ) + u ′ v ′ ( EG u − F E v ) + u ′ v ′ ( F G u − GE v )+ 12 u ′ v ′ ( EG v − F F v + F G u ) + 12 v ′ ( F G v − GF v + GG u ) . (3.21)In addition, let Γ kij be the Christoffel symbols of second kind given by(3.22) Γ = W { GE u + F [ E v − F u ] } , Γ = W { EG v + F [ G v − F v ] } Γ = W { E [2 F u − E v ] − F E v } , Γ = W { G [2 F v − G u ] − F G v } Γ = W { EG u − F E v } = Γ , Γ = W { GE v − F G u } = Γ , ONFORMAL IMAGE OF AN OSCULATING CURVE ON A SMOOTH IMMERSED SURFACE9 where W = √ EG − F . After conformal motion, the Christoffel symbols turns outto be ˜Γ = Γ + ϑ , ˜Γ = Γ + ϑ , ˜Γ = Γ + ϑ , ˜Γ = Γ + ϑ , ˜Γ = Γ + ϑ , ˜Γ = Γ + ϑ , (3.23)where ϑ = EGδ u − F δ u + F Eδ v δW , ϑ = EF δ u − E δ v δW ,ϑ = EGδ v − F Gδ u δW , ϑ = EGδ u − F Eδ v δW ,ϑ = GF δ v − G δ u δW , ϑ = EGδ v − F δ v + F Gδ u δW . (3.24)Using (3.22) in (3.21), we get(3.25) κ g = h Γ u ′ +(2Γ − Γ ) u ′ v ′ +(Γ − ) u ′ v ′ − Γ v ′ + u ′ v ′′ − u ′′ v ′ ip EG − F . In view of (3.23) and the above equation, ˜ κ g is given by(3.26) ˜ κ g = δ κ g + h ǫ u ′ +(2 ǫ − ǫ ) u ′ v ′ +( ǫ − ǫ ) u ′ v ′ − ǫ v ′ ip EG − F . This proves the claim. (cid:3)
Corollary 3.12.
Let G be homothetic conformal map between two smooth surfaces M and ˜ M . Then the geodesic curvature of rectifying curve is homothetic invariantunder G .Proof. Let us suppose δ = c, then the proof is a direct implication of (3.24) and(3.26). (cid:3) Corollary 3.13. [7]
Let G be an isometry between two smooth surfaces M and ˜ M .Then the geodesic curvature of rectifying curve is invariant under G . Conformal image of Geodesics
In this section, we seek what happens to a geodesic, in particular an osculatingcurve on a smooth immersed surface under conformal transformation.
Definition 4.1.
A vector field X is said to be parallel along a curve σ : I → M if DXdx = 0 for all x ∈ I. Definition 4.2.
A non-constant curve σ : I → M is said to be geodesic at x ∈ I if its tangent vector field is parallel at x ∈ I , i.e., Dσ ′ ( x ) dx = 0 and σ is said to be ageodesic on M if it is geodesic for all x ∈ M . Theorem 4.3.
Let G : M → ˜ M be a conformal map between two smooth surfaces M and ˜ M and σ ( s ) be a geodesic on M . Then ˜ σ = G ◦ σ being a geodesic on ˜ M is equivalent to the following system of differential equations (4.1) (cid:26) u ′′ + Γ u ′ + 2Γ u ′ v ′ + Γ v ′ + f ( E, F, G, δ ) = 0 v ′′ + Γ u ′ + 2Γ u ′ v ′ + Γ v ′ + f ( E, F, G, δ ) = 0 . Proof.
Let σ ( s ) = Φ( u ( s ) , v ( s )) be a parameterized geodesic on the surface M andΦ( u, v ) be the coordinate chart of M . Then, we have t ( s ) = σ ′ ( s ) = Φ u u ′ + Φ v v ′ . , MOHAMD SALEEM LONE AND PINAKI RANJAN GHOSH Taking the covariant derivative of the above expression, the parallel condition ofthe tangent vector field of σ is obtained as(4.2) (cid:16) u ′′ +Γ u ′ +2Γ u ′ v ′ +Γ v ′ (cid:17) Φ u + (cid:16) v ′′ +Γ u ′ +2Γ u ′ v ′ +Γ v ′ (cid:17) Φ v = 0 , where Γ kij , ( i, j, k = 1 ,
2) are Christoffel symbols given by (3.22).Since Φ u and Φ v are two basis vectors, therefore the geodesic condition in (4.2) isequivalent to the following system of differential equations:(4.3) (cid:26) u ′′ + Γ u ′ + 2Γ u ′ v ′ + Γ v ′ = 0 v ′′ + Γ u ′ + 2Γ u ′ v ′ + Γ v ′ = 0 . Let ˜ σ be a conformal image of σ on ˜ M . Using (3.23) and (3.24), we have(4.4) u ′′ + Γ u ′ + 2Γ u ′ v ′ + Γ v ′ + (cid:16) ϑ u ′ + 2 ϑ u ′ v ′ + ϑ v ′ (cid:17) = 0 v ′′ + Γ u ′ + 2Γ u ′ v ′ + Γ v ′ + (cid:16) ϑ u ′ + 2 ϑ u ′ v ′ + ϑ v ′ (cid:17) = 0 , where ǫ kij , ( i, j, k = 1 ,
2) are given by (3.24). Therefore the fact that ˜ σ is a geodesicon ˜ M is equivalent to the following system of differential equations: (cid:26) u ′′ + Γ u ′ + 2Γ u ′ v ′ + Γ v ′ + f ( E, F, G, δ ) = 0 v ′′ + Γ u ′ + 2Γ u ′ v ′ + Γ v ′ + f ( E, F, G, δ ) = 0 ,f ( E, F, G, δ ) = ϑ u ′ + 2 ϑ u ′ v ′ + ϑ v ′ and f ( E, F, G, δ ) = ϑ u ′ + 2 ϑ u ′ v ′ + ϑ v ′ . This proves the result. (cid:3) Corollary 4.4.
A geodesic say σ ( s ) on a smooth surface is invariant under isom-etry and homothetic motion.Proof. Noting that from (3.23) and (3.24), the Christofell symbols are invariant un-der isometry of homothetic motion. From (4.1), we see that the geodesic conditionsare dilated with the dilation factors f and f , but from (4.1), (3.23) and (3.24), itis easy to judge that the geodesics remain invariant under isometry and homotheticmotion. (cid:3) Note:
The conclusions in theorem 4.3 and corollary 4.4 are true for a generalspace curve, the same is true, in particular if σ ( s ) is an osculating curve.5. acknowledgment The second author greatly acknowledges to The University Grants Commission,Government of India for the award of Junior Research Fellow.
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