Normal curves on a smooth immersed surface
aa r X i v : . [ m a t h . G M ] J un NORMAL CURVES ON A SMOOTH IMMERSED SURFACE
ABSOS ALI SHAIKH , MOHAMD SALEEM LONE AND PINAKI RANJAN GHOSH Abstract.
The aim of this paper is to investigate the sufficient condition forthe invariance of a normal curve on a smooth immersed surface under isometry.We also find the the deviations of the tangential and normal components ofthe curve with respect to the given isometry. Introduction
When we talk of a manifold, one of the most elementary geometric object isits position vector field, which reveals the position of an arbitrary point on thatmanifold with respect to some origin. In case of curves, the position vector field ofa curve can be thought of as the motion of a particle with respect to a parameter s and intuitively the first and second derivatives of the curve gives the velocity andthe acceleration of the particle respectively. So, we shall discuss a problem whichaltogether depends upon the position vector field of a curve.Let α : I ⊂ R → E be a unit speed curve having all the necessary propertiessuch that { t, n, b } acts as its Serret-Frenet frame, where t, n , and b are the tangent,the normal and the binormal vectors(unitary), respectively. Then, the Serret-Frenetequations are given by t ′ = κnn ′ = − κt + τ bb ′ = − τ n, (1.1)where κ is the curvature and τ is the torsion of α with t = α ′ , n = t ′ κ , b = t × n ,and ′ denotes the differentiation with respect to the parameter s . At each point α ( s ) of α , the planes spanned by { t, n } , { t, b } and { n, b } are called as the osculatingplane, the rectifying plane and the normal plane, respectively. As it is evident fromthe names of planes, a curve whose position vector field lies in the osculating planeis called as an osculating curve. Similarly, a curve whose position vector filed liesin the rectifying and the normal plane are called as rectifying and normal curves,respectively.It is well known that if at each point the position vector of α lies in the osculatingplane, then the curve lies in a plane. Similarly, if the the position vector of α liesin the normal plane at each point, then the curve lies on a sphere. So, in view ofthese facts, in 2003 Chen([3]) posed a question: When does the position vector of acurve lies in the rectifying curve? Therein, Chen obtained characterization resultsfor rectifying curves. In relation to these three types of curves, enormous study hasbeen done. For generic study, we refer the reader to see([4, 5, 6, 7]). Mathematics Subject Classification.
Key words and phrases.
Isometry, normal curve, osculating curve, rectifying curve. , MOHAMD SALEEM LONE AND PINAKI RANJAN GHOSH Before going to motivation and objective of this paper, we revisit a few definitions([2]) on surfaces.
Definition 1.1.
A diffeomorphism J : M → M is an isometry if for all p ∈ M andall pairs x , x ∈ T p ( M ), we have h x , x i p = h dJ p ( x ) , dJ p ( x ) i J ( p ) . The surfaces M and M are then said to be isometric. Definition 1.2.
A map J : V ⊂ M → M of a neighborhood V of p ∈ M is a localisometry at p if there exists a neighborhood U of J ( p ) ∈ M such that J : V → U is an isometry. If there exists local isometry at every point of M , then the surfaces M and M are said to be locally isometric. Clearly, if J is a diffeomorphism and alocal isometry for every p ∈ M , then J is global isometry.It is straightforward to see that the first fundamental form coefficients are pre-served under isometry. So if E, F, G and
E, F , G are the first fundamental formcoefficients of M and M , respectively and J : M → M is a local isometry then E = E, F = F , G = G. In 2018 Shaikh and Ghosh ([8]) diverted the study of rectifying curves to a newdirection by questioning about the invariant properties of a rectifying curve on asmooth surface under isometry. In addition to a sufficient condition for a rectifyingcurve to remain invariant under isometry, they showed that the component of therectifying curve along the surface normal is invariant under isometry. Again in ([9])Shaikh and Ghosh studied osculating curves and obtained their characterizationalong with invariancy under surface isometry. Motivated by ([8], [9] and [10]), weshall investigate the similar questions in case of normal curves , i.e.,
Question:
What happens to a normal curve on a smooth surface under isometry?In the section 2, we give some of the basic notions about normal curves and findthe Frenet frame vectors of the normal curves with respect to the smooth immersedsurface. Section 3 is concerned with the main results and provided the answer ofabove question. 2.
Preliminaries
Let α be a normal curve parameterized by arc with a Serret-Frenet frame givenin (1.1). The other way of interpreting a normal curve is: a curve is said to bea normal curve if its position vector lies in the orthogonal complement of tangentvector i.e., α · t = 0 , or(2.1) α ( s ) = λ ( s ) n ( s ) + µ ( s ) b ( s ) , where λ, µ are two smooth functions.Suppose M is a regular surface(page no 52, [2]) with ϕ ( u, v ) being its coordinatechart. Then, the curve α ( s ) = α ( u ( s ) , v ( s )) defines a curve α ( s ) = M ( u ( s ) , v ( s ))on the surface M . We can easily find the derivatives of the curve α ( s ) as a curveon the surface M using the chain rule: α ′ ( s ) = ϕ u u ′ + ϕ v v ′ or, t ( s ) = α ′ ( 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 . ORMAL CURVES ON A SMOOTH IMMERSED SURFACE 3
Now let N be the unit surface normal then we have n ( s ) = 1 k ( s ) ( u ′′ ϕ u + v ′′ ϕ v + u ′ ϕ uu + 2 u ′ v ′ ϕ uv + v ′ ϕ vv ) .b ( s ) = t ( s ) × n ( s ) = t ( s ) × t ′ ( s ) k ( s )= 1 k ( s ) h ( ϕ u u ′ + ϕ v v ′ ) × ( u ′′ ϕ u + v ′′ ϕ v + u ′ ϕ uu + 2 u ′ v ′ ϕ uv + v ′ ϕ vv ) i = 1 k ( s ) h { u ′ v ′′ − u ′′ v ′ } N + u ′ ϕ u × ϕ uu + 2 u ′ v ′ ϕ u × ϕ uv + u ′ v ′ ϕ u × ϕ vv + u ′ v ′ ϕ v × ϕ uu + 2 u ′ v ′ ϕ v × ϕ uv + v ′ ϕ v × ϕ vv i . (2.3) Definition 2.1. [1] Let α be a unit speed curve on M , then the unit tangent vector t = α ′ is orthogonal to the unit surface normal N , so α ′ , N and N × α ′ are mutuallyorthogonal vectors. Moreover, since α ′′ is orthogonal to α ′ , we can write α ′′ as alinear combination of N and N × α ′ , i.e., α ′′ = κ n N + κ g N × α ′ , where κ n and κ g are called as the normal curvature and the geodesic curvature of α , respectively and are given by (cid:26) κ g = α ′′ · N × α ′ κ n = α ′′ · N . Now since α ′′ = κ ( s ) n ( s ), therefore we can write 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 second fundamental form coefficients of the surface. Thecurve α on M is said to be asymptotic if and only if κ n = 0 . Normal curves
The equation of a normal curve is given by(3.1) α ( s ) = λ ( s ) n ( s ) + µ ( s ) b ( s ) . Suppose this curve lies on a parametric surface ϕ ( u, v ). Then (3.1) is in the form: α ( s ) = λ ( s ) κ ( s ) h ( u ′′ ϕ u + v ′′ ϕ v ) + ( u ′ ϕ uu + 2 u ′ v ′ ϕ uv + v ′ ϕ vv ) i + µ ( s ) k ( s ) h { u ′ v ′′ − u ′′ v ′ } ~N + u ′ ϕ u × ϕ uu + 2 u ′ v ′ ϕ u × ϕ uv (3.2) + u ′ v ′ ϕ u × ϕ vv + u ′ v ′ ϕ v × ϕ uu + 2 u ′ v ′ ϕ v × ϕ uv + v ′ ϕ v × ϕ vv i . Theorem 3.1.
Let M and M be two smooth surfaces and J : M → M be anisometry. Also, let α ( s ) be a normal curve on M . Then α ( s ) = J ◦ α ( s ) is a ABSOS ALI SHAIKH , MOHAMD SALEEM LONE AND PINAKI RANJAN GHOSH normal curve on M if α ( s ) − J ∗ ( α ( s )) = λ ( s ) κ ( s ) h u ′ (cid:18) ∂J ∗ ∂u ϕ u (cid:19) + 2 u ′ v ′ (cid:18) ∂J ∗ ∂u ϕ v (cid:19) + v ′ (cid:18) ∂J ∗ ∂v ϕ v (cid:19) i + µ ( s ) k ( s ) h u ′ (cid:16) J ∗ ϕ u × ∂J ∗ ∂u ϕ u (cid:17) + 2 u ′ v ′ (cid:16) J ∗ ϕ u × ∂J ∗ ∂u ϕ v (cid:17) (3.3) + u ′ v ′ (cid:16) J ∗ ϕ u × ∂J ∗ ∂v ϕ v (cid:17) + u ′ v ′ (cid:16) J ∗ ϕ v × ∂J ∗ ∂u ϕ u (cid:17) +2 u ′ v ′ (cid:16) J ∗ ϕ v × ∂J ∗ ∂u ϕ v (cid:17) + v ′ (cid:16) J ∗ ϕ v × ∂J ∗ ∂v ϕ v (cid:17)i . Proof.
Suppose ϕ and ϕ are the chart maps of M and M , respectively. Then, wehave ϕ = J ◦ ϕ.J : M → M is an isometry whose differential map dJ = J ∗ is a 3 × T p ( M ) to linearly independent vectorsof T J ( p ) M , i.e., J ∗ : T p ( M ) → T J ( p ) M . Since { ϕ u , ϕ v } is a basis of tangent plane T p ( M ) at a point p on M , we have¯ ϕ u ( u, v ) = J ∗ ϕ u = J ∗ ( ϕ ( u, v )) ϕ u , (3.4) ¯ ϕ v ( u, v ) = J ∗ ϕ v = J ∗ ( ϕ ( u, v )) ϕ v . (3.5)Differentiating (3.4) and (3.5) with respect to u, v , we get¯ ϕ uu = ∂J ∗ ∂u ϕ u + J ∗ ϕ uu , ¯ ϕ vv = ∂J ∗ ∂v ϕ v + J ∗ ϕ vv , (3.6) ¯ ϕ uv = ∂J ∗ ∂u ϕ v + J ∗ ϕ uv = ∂J ∗ ∂v ϕ u + J ∗ ϕ uv . We can write(3.7) J ∗ ϕ u × ∂J ∗ ∂u ϕ u = J ∗ ϕ u × (cid:16) ∂J ∗ ∂u ϕ u + F ∗ ϕ uu (cid:17) − J ∗ ( ϕ u × ϕ uu ) = ¯ ϕ u × ¯ ϕ uu − J ∗ ( ϕ u × ϕ uu ) . Similarly J ∗ ϕ u × ∂J ∗ ∂u ϕ v = ¯ ϕ u × ¯ ϕ uv − J ∗ ( ϕ u × ϕ uv ) ,J ∗ ϕ u × ∂J ∗ ∂v ϕ v = ¯ ϕ u × ¯ ϕ vv − J ∗ ( ϕ u × ϕ vv ) ,J ∗ ϕ v × ∂J ∗ ∂u ϕ u = ¯ ϕ v × ¯ ϕ uu − J ∗ ( ϕ v × ϕ uu ) , (3.8) J ∗ ϕ v × ∂J ∗ ∂u ϕ v = ¯ ϕ v × ¯ ϕ uv − J ∗ ( ϕ v × ϕ uv ) ,J ∗ ϕ v × ∂J ∗ ∂v ϕ v = ¯ ϕ v × ¯ ϕ vv − J ∗ ( ϕ v × ϕ vv ) . ORMAL CURVES ON A SMOOTH IMMERSED SURFACE 5
Therefore, with respect to (3 . α ( s ) = λ ( s ) κ ( s ) h u ′′ J ∗ ϕ u + v ′′ J ∗ ϕ v + u ′ (cid:18) ∂J ∗ ∂ u ϕ u + J ∗ ϕ uu (cid:19) + 2 u ′ v ′ (cid:18) ∂J ∗ ∂u ϕ v + J ∗ ϕ uv (cid:19) + v ′ (cid:18) ∂J ∗ ∂v ϕ v + J ∗ ϕ vv (cid:19) i + µ ( s ) k ( s ) h { u ′ v ′′ − u ′′ v ′ } J ∗ ~N + u ′ J ∗ ( ϕ u × ϕ uu )+2 u ′ v ′ J ∗ ( ϕ u × ϕ uv ) + u ′ v ′ J ∗ ( ϕ u × ϕ vv ) + u ′ v ′ J ∗ ( ϕ v × ϕ uu ) + 2 u ′ v ′ J ∗ ( ϕ v × ϕ uv )+ v ′ J ∗ ( ϕ v × ϕ vv ) + u ′ (cid:16) J ∗ ϕ u × ∂J ∗ ∂u ϕ u (cid:17) + 2 u ′ v ′ (cid:16) J ∗ ϕ u × ∂J ∗ ∂u ϕ v (cid:17) + u ′ v ′ (cid:16) J ∗ ϕ u × ∂J ∗ ∂v ϕ v (cid:17) + u ′ v ′ (cid:16) J ∗ ϕ v × ∂J ∗ ∂u ϕ u (cid:17) + 2 u ′ v ′ (cid:16) J ∗ ϕ v × ∂J ∗ ∂u ϕ v (cid:17) + v ′ (cid:16) J ∗ ϕ v × ∂J ∗ ∂v ϕ v (cid:17)i , or α ( s ) = λ ( s ) κ ( s ) h ( u ′′ ϕ u + v ′′ ϕ v ) + ( u ′ ϕ uu + 2 u ′ v ′ ϕ uv v ′ ϕ vv ) i + µ ( s ) k ( s ) h { u ′ v ′′ − u ′′ v ′ } ~N + u ′ ϕ u × ϕ uu + 2 u ′ v ′ ϕ u × ϕ uv (3.9) + u ′ v ′ ϕ u × ϕ vv + u ′ v ′ ϕ v × ϕ uu + 2 u ′ v ′ ϕ v × ϕ uv + v ′ ϕ v × ϕ vv i . Therefore(3.10) α ( s ) = λ ( s ) κ ( s ) n ( s ) + µ ( s ) κ ( s ) b ( s )for some functions λ ( s ) and µ ( s ). Therefore, α ( s ) is a normal curve in M . (cid:3) Theorem 3.2.
Let J : M → M be an isometry and α ( s ) be a normal curve on M .Then for the tangential components, we have ¯ α · ¯T − α · T = µκ (¯ κ n − κ n )( av ′ + bu ′ ) , (3.11) where T = aϕ u + bϕ v is any tangent vector to M for some a, b ∈ R .Proof. From (3.2), we see that α · ϕ u = λκ h u ′′ E + v ′′ F + u ′ ϕ uu · ϕ u + 2 u ′ v ′ ϕ uv · ϕ u + v ′ ϕ vv · ϕ u i + µκ h u ′ v ′ L + 2 v ′ u ′ M + v ′ N i . (3.12)Now for the isometric images of α and ϕ u , we have α · ϕ u = λκ h u ′′ E + v ′′ F + u ′ ϕ uu · ϕ u + 2 u ′ v ′ ϕ uv · ϕ u + v ′ ϕ vv · ϕ u i + µκ h u ′ v ′ L + 2 v ′ u ′ M + v ′ N i . (3.13)Since we know that E = E, F = F, G = G . In particular E = E = J ∗ ϕ u · J ∗ ϕ u = ϕ u · ϕ u Differentiating the above equation with respect to u , we get (cid:18) ∂J ∗ ∂u ϕ u + J ∗ ϕ uu (cid:19) · ( J ∗ ϕ u ) = ϕ uu · ϕ u ABSOS ALI SHAIKH , MOHAMD SALEEM LONE AND PINAKI RANJAN GHOSH or ϕ uu · ϕ u = ϕ uu · ϕ u . Similarly, it is easy to show that ϕ uv · ϕ u = ϕ uv · ϕ u , ϕ vv · ϕ u = ϕ vv · ϕ u . Thus from (3.13), we get α · ϕ u = λκ h u ′′ E + v ′′ F + u ′ ϕ uu · ϕ u + 2 u ′ v ′ ϕ uv · ϕ u + v ′ ϕ vv · ϕ u i + µκ h u ′ v ′ L + 2 v ′ u ′ M + v ′ N i , or(3.14) α · ϕ u = λκ h u ′′ E + v ′′ F + u ′ ϕ uu · ϕ u + 2 u ′ v ′ ϕ uv · ϕ u + v ′ ϕ vv · ϕ u i + v ′ µκ κ n . Taking the difference of (3.14) and (3.12), we get(3.15) α · ϕ u − α · ϕ u = v ′ µκ ( κ n − κ n ) . Similarly the following relation hold(3.16) α · ϕ v − α · ϕ v = u ′ µκ ( κ n − κ n ) . Now with the help of (3 .
15) and (3 .
16) we get¯ α · ¯T − α · T = ¯ α · ( a ¯ ϕ u + b ¯ ϕ v ) − α · ( aϕ u + bϕ v )= a ( α · ϕ u − α · ϕ u ) + b ( α · ϕ v − α · ϕ v )= av ′ µκ ( κ n − κ n ) + bu ′ µκ ( κ n − κ n )= µκ (¯ κ n − κ n )( av ′ + bu ′ ) . This proves our claim. (cid:3)
Corollary 3.3.
Let J : M → M be an isometry and α ( s ) be a normal curve on M . Then the component of the normal curve α ( s ) along any tangent vector T tothe surface M is invariant if and only if any one of the following holds: (i) the position vector of α ( s ) is in the normal direction of α . (ii) The normal curvature is invariant.Proof.
From (3 . α · ¯T = α · T if and only if µκ (¯ κ n − κ n )( av ′ + bu ′ ) = 0i.e., if and only if µ = 0 or ¯ κ n − κ n = 0 . If µ = 0 then from (3 . α ( s ) = λ ( s ) n ( s ), i.e., the position vector ofthe normal curve α ( s ) is in the normal direction of itself. (cid:3) Corollary 3.4.
Let J : M → M be an isometry and α ( s ) be a normal curve on M .The component of the normal curve α ( s ) along any tangent vector T to the surface M is invariant and the position vector of α ( s ) is not in the normal direction of α ,then α ( s ) is asymptotic if and only if ¯ α ( s ) is asymptotic. ORMAL CURVES ON A SMOOTH IMMERSED SURFACE 7
Proof.
From Corollary 3 . . , ¯ α · ¯T = α · T and the position vector of α ( s ) is not inthe normal direction of α if and only if κ n = ¯ κ n .Therefore α ( s ) is asymptotic if and only if κ n = 0 if and only if ¯ κ n = 0 if and onlyif ¯ α ( s ) is asymptotic. (cid:3) Theorem 3.5.
Let J : M → M be an isometry and α ( s ) be a normal curve on M .Then for the component of α ( s ) along the surface normal N , we have (3.17) α · N − α · N = λκ ( κ n − κ n ) . Proof.
From (3.2), we have α · N = λκ h u ′ ϕ uu · ( ϕ u × ϕ v ) + v ′ ϕ vv · ( ϕ u × ϕ v ) + 2 u ′ v ′ ϕ uv · ( ϕ u × ϕ v ) i + µκ h ( u ′ v ′′ − v ′ u ′′ )( EG − F ) + u ′ ( ϕ u × ϕ uu ) · ( ϕ u × ϕ v )+2 u ′ v ′ ( ϕ u × ϕ uv ) · ( ϕ u × ϕ v ) + v ′ u ′ ( ϕ u × ϕ vv ) · ( ϕ u × ϕ v )+ u ′ v ′ ( ϕ v × ϕ uu ) · ( ϕ u × ϕ v ) + 2 u ′ v ′ ( ϕ v × ϕ uv ) · ( ϕ u × ϕ v )+ v ′ ( ϕ v × ϕ vv ) · ( ϕ u × ϕ v ) i or α · N = λκ h u ′ L + v ′ N + 2 u ′ v ′ M i + µκ h ( u ′ v ′′ − v ′ u ′′ )( EG − F ) + u ′ { E ( ϕ uu · ϕ v ) − F ( ϕ uu · ϕ u ) } + 2 u ′ v ′ { E ( ϕ uv · ϕ v ) − F ( ϕ uv · ϕ u ) } + u ′ v ′ { E ( ϕ vv · ϕ v ) − F ( ϕ vv · ϕ u ) } + u ′ v ′ { F ( ϕ uu · ϕ v ) − G ( ϕ uu · ϕ u ) } + 2 u ′ v ′ { F ( ϕ uv · ϕ v ) − G ( ϕ uv · ϕ u ) } + v ′ { F ( ϕ vv · ϕ v ) − G ( ϕ vv · ϕ u ) } i . Since we know that with respect to isometry: E = E, F = F , G = G . Then, it easyto verify: (cid:26) E u = E u , F u = F u , G u = G u E v = E v , F v = F v , G v = G v . (3.18)Then we have E u = ( ϕ u · ϕ u ) u = ϕ uu · ϕ u or(3.19) ϕ uu · ϕ u = E u . On the similar lines, we can find (cid:26) ϕ uu · ϕ v = F u − E v , ϕ vv · ϕ v = G v , ϕ vv · ϕ u = F v = G u ϕ uv · ϕ v = G u , ϕ uv · ϕ u = E v (3.20) ABSOS ALI SHAIKH , MOHAMD SALEEM LONE AND PINAKI RANJAN GHOSH Therefore in view of (3.19) and (3.20), we get α · N = λκ h u ′ L + v ′ N + 2 u ′ v ′ M i + µκ h ( u ′ v ′′ − v ′ u ′′ )( EG − F ) + u ′ (cid:26) E (cid:18) F u − E v (cid:19) − F E u (cid:27) +2 u ′ v ′ (cid:26) EG u − F E v (cid:27) + u ′ v ′ (cid:26) EG v − F (cid:18) F v − G u (cid:19)(cid:27) + u ′ v ′ (cid:26) F G u − GE u (cid:27) + 2 u ′ v ′ (cid:26) F G u − GE v (cid:27) + v ′ (cid:26) F G v − G (cid:18) F v − G u (cid:19)(cid:27) i . Now applying J and with the help of (3.18), we get α · N = λκ h u ′ L + v ′ N + 2 u ′ v ′ M i + µκ h ( u ′ v ′′ − v ′ u ′′ )( EG − F ) + u ′ (cid:26) E (cid:18) F u − E v (cid:19) − F E u (cid:27) +2 u ′ v ′ (cid:26) EG u − F E v (cid:27) + u ′ v ′ (cid:26) EG v − F (cid:18) F v − G u (cid:19)(cid:27) + u ′ v ′ (cid:26) F G u − GE u (cid:27) + 2 u ′ v ′ (cid:26) F G u − GE v (cid:27) + v ′ (cid:26) F G v − G (cid:18) F v − G u (cid:19)(cid:27) i . On taking the difference of α · N and its isometric image and with the help of (3 . α · N − α · N = λκ ( κ n − κ n ) . This proves our claim. (cid:3)
Corollary 3.6.
Let J : M → M be an isometry and α ( s ) be a normal curve on M .Then the component of the normal curve α ( s ) along the surface normal is invariantif and only if any one of the following holds: (i) The position vector of α ( s ) is in the binormal direction of α . (ii) The normal curvature is invariant.Proof.
From (3 . α · ¯N = α · N if and only if λκ ( κ n − κ n ) = 0i.e., if and only if λ = 0 or κ n − κ n = 0 . If λ = 0 then from (3 . α ( s ) = µ ( s ) b ( s ), i.e., the position vector ofthe normal curve α ( s ) is in the binormal direction of itself. (cid:3) Corollary 3.7.
Let J : M → M be an isometry and α ( s ) be a normal curve on M .The component of the normal curve α ( s ) along surface normal N to the surface M is invariant and the position vector of α ( s ) is not in the normal direction of α , then α ( s ) is asymptotic if and only if ¯ α ( s ) is asymptotic. ORMAL CURVES ON A SMOOTH IMMERSED SURFACE 9
Proof.
From Corollary 3 . . , ¯ α · ¯N = α · N and the position vector of α ( s ) is not inthe normal direction of α if and only if κ n = ¯ κ n .Therefore α ( s ) is asymptotic if and only if κ n = 0 if and only if ¯ κ n = 0 if and onlyif ¯ α ( s ) is asymptotic. (cid:3) Since T and N are perpendicular vector at α ( s ), hence { T , N , T × N } form anorthogonal system at every point of the normal curve α ( s ). Theorem 3.8.
Let J : M → M be an isometry and α ( s ) be a normal curve on M .Then for the component of α ( s ) along T × N , we have (3.21) ¯ α · ( ¯T × ¯N ) − α · ( T × N ) = µκ (¯ κ n − κ n ) { a ( F v ′ − Eu ′ ) + b ( Gv ′ − F u ′ ) } Proof.
From (2 . α · ( T × N ) = α · { ( aϕ u + bϕ v ) × N } , = α · { a ( F ϕ u − Eϕ v ) + b ( Gϕ u − F ϕ v ) } , = ( aF + bG ) α · ϕ u − ( aE + bF ) α · ϕ v . Therefore using (3 .
15) and (3 .
16) we get¯ α · ( ¯ T × ¯ N ) − α · ( T × N ) = ( aF + bG )( α · ϕ u − α · ϕ u ) − ( aE + bF )( α · ϕ v − α · ϕ v ) , = µκ (¯ κ n − κ n ) { a ( F v ′ − Eu ′ ) + b ( Gv ′ − F u ′ ) } . This proves our claim. (cid:3)
Corollary 3.9.
Let J : M → M be an isometry and α ( s ) be a normal curve on M . Then the component of the normal curve α ( s ) along T × N is invariant if andonly if any one of the following holds: (i) The position vector of α ( s ) is in the normal direction of α . (ii) The normal curvature is invariant.Proof.
From (3 . α · ( ¯ T × ¯ N ) = α · ( T × N ) if and only if µκ (¯ κ n − κ n ) { a ( F v ′ − Eu ′ ) + b ( Gv ′ − F u ′ ) } = 0i.e., if and only if µ = 0 or ¯ κ n − κ n = 0 . If µ = 0 then from (3 . α ( s ) = λ ( s ) n ( s ), i.e., the position vector ofthe normal curve α ( s ) is in the normal direction of itself. (cid:3) Corollary 3.10.
Let J : M → M be an isometry and α ( s ) be a normal curve on M . The component of the normal curve α ( s ) along T × N is invariant and theposition vector of α ( s ) is not in the normal direction of α , then α ( s ) is asymptoticif and only if ¯ α ( s ) is asymptotic.Proof. From Corollary 3 . . , ¯ α · ( ¯ T × ¯ N ) = α · ( T × N ) and the position vector of α ( s ) is not in the normal direction of α if and only if κ n = ¯ κ n .Therefore α ( s ) is asymptotic if and only if κ n = 0 if and only if ¯ κ n = 0 if and onlyif ¯ α ( s ) is asymptotic. (cid:3) Proposition 1.
The geodesic curvature of a smooth curve and in particular of anormal curve remains invariant under isometry. , MOHAMD SALEEM LONE AND PINAKI RANJAN GHOSH Proof.
Let α be a curve on a parametric surface M , then the geodesic curvature isgiven by Beltrami formula as:(3.22) κ g = h Γ u ′ +(2Γ − Γ ) u ′ v ′ +(Γ − ) u ′ v ′ − Γ v ′ + u ′ v ′′ − u ′′ v ′ ip EG − F , where Γ kij are the Christoffel symbols of the second kind given by(3.23) 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 and W = √ EG − F . Thus, in view of (3.18), (3.22) and (3.23), we see that κ g = κ g . In particular the same holds for a normal curve. (cid:3) acknowledgment The third author greatly acknowledges to The University Grants Commission,Government of India for the award of Junior Research Fellow.
References [1] A. Pressley,
Elementary differential geometry , Springer-Verlag, 2001.[2] M. P. do Carmo,
Differential geometry of curves and surfaces , Prentice-Hall, Inc, New Jersey,1976.[3] B.-Y. Chen,
What does the position vector of a space curve always lie in its rectifying plane ?,Amer. Math. Monthly, (2003), 147-152.[4] B.-Y. Chen and F. Dillen,
Rectfying curve as centrode and extremal curve. , Bull. Inst. Math.Acad. Sinica, , no. 2, (2005), 77-90.[5] S. Deshmukh, B.-Y. Chen and S. H. Alshammari, On a rectifying curves in euclidean 3-space ,Turk. J. Math., (2018), 609-620.[6] M. Grbovi´c and Emilija Neˇsovi´c, Some relations between rectifying and normal curves inMinkowski 3-space , Math. Commun., (2012), 655-664.[7] K. Ilarslan, M. Sakaki and A. U¸cum, On osculating, normal and rectifying bi-null curves in R , Novi Sad J. Math. , (2018), 9-20.[8] A. A Shaikh and P. R. Ghosh Rectifying curves on a smooth surface immersed in the Euclideanspace , to appear in Indian J. Pure Appl. Math., (2018).[9] A. A Shaikh and P. R. Ghosh
Rectifying and osculating curves on a smooth surface , to appearin Indian J. Pure Appl. Math., (2018).[10] A. A Shaikh and P. R. Ghosh
Curves on a smooth surface with position vectors lie in thetangent plane , Submited, (2018). Department of Mathematics, University of Burdwan, Golapbag, Burdwan-713104,West Bengal, India
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