Ruled Surfaces in E⁴ with Constant Ratio of the Gaussian Curvature and Gaussian Torsion

Ruled Surfaces in E⁴ with Constant Ratio of the Gaussian Curvature and Gaussian Torsion

Local and global existence theorems on the ruled surfaces with a constant ratio of the Gaussian curvature and Gaussian torsion are proved.

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Datum:2008
1. Verfasser: Goncharova, O.A.
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Sprache:English
Veröffentlicht: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2008
Schriftenreihe:Журнал математической физики, анализа, геометрии
Online Zugang:http://dspace.nbuv.gov.ua/handle/123456789/106513
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Zitieren:Ruled Surfaces in E⁴ with Constant Ratio of the Gaussian Curvature and Gaussian Torsion / O.A. Goncharova // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 3. — С. 371-379. — Бібліогр.: 4 назв. — англ.

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spelling irk-123456789-1065132016-09-30T03:02:58Z Ruled Surfaces in E⁴ with Constant Ratio of the Gaussian Curvature and Gaussian Torsion Goncharova, O.A. Local and global existence theorems on the ruled surfaces with a constant ratio of the Gaussian curvature and Gaussian torsion are proved. 2008 Article Ruled Surfaces in E⁴ with Constant Ratio of the Gaussian Curvature and Gaussian Torsion / O.A. Goncharova // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 3. — С. 371-379. — Бібліогр.: 4 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106513 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description Local and global existence theorems on the ruled surfaces with a constant ratio of the Gaussian curvature and Gaussian torsion are proved.
format Article
author Goncharova, O.A.
spellingShingle Goncharova, O.A.
Ruled Surfaces in E⁴ with Constant Ratio of the Gaussian Curvature and Gaussian Torsion
Журнал математической физики, анализа, геометрии
author_facet Goncharova, O.A.
author_sort Goncharova, O.A.
title Ruled Surfaces in E⁴ with Constant Ratio of the Gaussian Curvature and Gaussian Torsion
title_short Ruled Surfaces in E⁴ with Constant Ratio of the Gaussian Curvature and Gaussian Torsion
title_full Ruled Surfaces in E⁴ with Constant Ratio of the Gaussian Curvature and Gaussian Torsion
title_fullStr Ruled Surfaces in E⁴ with Constant Ratio of the Gaussian Curvature and Gaussian Torsion
title_full_unstemmed Ruled Surfaces in E⁴ with Constant Ratio of the Gaussian Curvature and Gaussian Torsion
title_sort ruled surfaces in e⁴ with constant ratio of the gaussian curvature and gaussian torsion
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2008
url http://dspace.nbuv.gov.ua/handle/123456789/106513
citation_txt Ruled Surfaces in E⁴ with Constant Ratio of the Gaussian Curvature and Gaussian Torsion / O.A. Goncharova // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 3. — С. 371-379. — Бібліогр.: 4 назв. — англ.
series Журнал математической физики, анализа, геометрии
work_keys_str_mv AT goncharovaoa ruledsurfacesine4withconstantratioofthegaussiancurvatureandgaussiantorsion
first_indexed 2025-07-07T18:35:29Z
last_indexed 2025-07-07T18:35:29Z
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fulltext Journal of Mathematical Physics, Analysis, Geometry 2008, vol. 4, No. 3, pp. 371�379 Ruled Surfaces in E 4 with Constant Ratio of the Gaussian Curvature and Gaussian Torsion O.A. Goncharova B. Verkin Institute for Low Temperature Physics and Engineering National Academy of Sciences of Ukraine 47 Lenin Ave., Kharkiv, 61103, Ukraine E-mail:goncharova@ilt.kharkov.ua Received November 3, 2006 Local and global existence theorems on the ruled surfaces with a constant ratio of the Gaussian curvature and Gaussian torsion are proved. Key words: ruled surfaces, Gaussian curvature, Gaussian torsion. Mathematics Subject Classi�cation 2000: 53A07. For 2-dimensional surfaces in E4 the Gaussian torsion is the invariant of nor- mal connection similar to the invariant of tangent connection, that is to the Gaus- sian curvature. Two-dimensional surfaces in E4 for which the Gaussian torsion �� coincides with the Gaussian curvature K were considered in [1]. For example, if the position vector of a surface is x; y; u(x; y); v(x; y) and u = �x, v = �y, where � = �(x; y) is some function, then K = ��. In general case on the surface with K = �� the functions u and v satisfy some di�erential equation of the �rst order. By using a method of characteristics one can integrate this equation to the system of Hamilton equations. Thus, in [1] the method of constructing the surfaces mentioned above is proposed and concrete examples are considered. It is natural to consider a wider class of surfaces for which the ratio of the Gaussian torsion and Gaussian curvature is a constant number. In this paper a class of the ruled surfaces in E4 is considered. It is assumed that the ratio of the Gaussian torsion and curvature is distinct from zero, as the ruled surfaces with zero Gaussian torsion were considered in [2]. In this paper from [2] the formulas for K and �� of the ruled surfaces in E4 are used. We formulate the basic results. Theorem 1. For any C2-regular curve � E4, with the �rst curvature k1 6= 0, and for any number c1 6= 0 in a neighborhood of each point p0 2 , there exists a regular ruled surface with the base curve for which the ratio of the Gaussian torsion and curvature is constant and equal to c1. c O.A. Goncharova, 2008 O.A. Goncharova Following Prop. A and B state global resolvability. Analogously to the slope curve in E3, denote the slope curve in E4 for which the curvatures k1 6= 0, k2 and k3 satisfy the relations k2 = �k1 and k3 = �k1, where �, � are constants, and � 6= 0. A generalized standard ruled surface is denoted as the two-dimensional ruled surface F 2 in Euclidean space En when the directing (base) curve is a curve with the distinct from zero curvatures k1; : : : ; kn�1, and the generatrix in each point are directed as one of the basis vectors from the natural frame �1; : : : ; �n. If the directing curve is a curve with constant and distinct from zero curvatures, then the generalized standard ruled surface corresponds to the standard ruled surface given in [4]. Proposition A. If in E4 is a slope curve, then there exists a regular gener- alized standard ruled surface along the whole base curve for which the ratio �� K is constant and equal to � �2+�2 . In particular, the standard ruled surface �3 from [4] is of this kind, what follows from the expressions for K and �� calculated in [4]. Proposition B. If is a �at curve with (k1 6= 0), then in E4 there exists a regular ruled surface along the whole base curve for which �� K = c1 = const. If the curve is a closed �at curve with the length `, and c1 = 1 2� R̀ 0 k1(t)dt, then the ruled surface is homeomorphic to a cylinder. As a special case of the closed �at curve can be taken a circle. If c1 = p q is a rational number and the curve is a circle taken p times, then this surface is homeomorphic to a cylinder. If c1 is an irrational number and the curve is a circle taken in�nite number of times, then this surface is homeomorphic to a plane. Some de�nitions should be recalled. Let be some Ck(k � 2)-regular curve in E4 with the position vector �(t), where t is an arc length. Let along the curve the Ck(k � 2)-regular unit vector �eld of generatrix a(t) be given. Then the position vector of the ruled surfaces F 2 has the following form: r(t; u) = �(t) + ua(t): Denote the curve as a base curve and also assume that a base curve is orthogonal to generatrix. Take the unit vector �eld a(t) as a linear combination of vectors of Frene's na- tural basis �2, �3, �4, that is a(t) = 4P i=2 ai�i. The expression for the �rst derivative of vector a has the form a0 = 4P i=1 T i�i, where the coe�cients T i are: T 1 = �k1a 2; T 2 = da2 dt � k2a 3; 372 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 Ruled Surfaces in E4 with Constant Ratio of the Gaussian Curvature... T 3 = da3 dt + k2a 2 � k3a 4; T 4 = da4 dt + k3a 3: From [2] the expression for the ratio of the Gaussian torsion and Gaussian curvature of the ruled surface in E4 is taken �� K = c(t) + b(t)u d(t) ; where c(t) = k1 �� a4 da3 dt � a3 da4 dt � � k3[(a 3)2 + (a4)2] + k2a 2a4 � ; b(t) = � T 4dT 3 dt � T 3 dT 4 dt � a2 + T 2 � dT 4 dt a3 � dT 3 dt a4 � +T 2T 4(k2a 2 + k3a 4)� (T 2)2k2a 4 + T 2T 3k3a 3 � k3a 2((T 3)2 + (T 4)2) +(T 3a4 � T 4a3) � dT 2 dt + k1T 1 � k2T 3 � ; d(t) = �(a0)2 + (�; a0)2 = �[(T 2)2 + (T 3)2 + (T 4)2]: Let us prove Th. 1. For this we consider the case �� K = c1 = const. To �nd the surfaces possessing this property it is necessary to solve the system b(t) = 0; (1) c(t) = c1d(t): (2) First, consider the equation (1). By substituting the expressions for T i, one gets d2a2 dt2 � a4 da3 dt � a3 da4 dt + k2a 2a4 � k3((a 3)2 + (a4)2) � + d2a3 dt2 � a2 da4 dt � a4 da2 dt + k3a 2a3 + k2a 3a4 � + d2a4 dt2 � a3 da2 dt � a2 da3 dt + k3a 2a4 � k2((a 2)2 + (a3)2) � + : : : = 0; (3) hereinafter the dots designate the terms not containing the second derivative from ai. Rewrite (2) in the following form: k1(T 3a4 � T 4a3) + c1((T 2)2 + (T 3)2 + (T 4)2) = 0: (4) By di�erentiating both parts of (4), one obtains k1 � dT 3 dt a4 � dT 4 dt a3 + : : : � + 2c1 � T 2dT 2 dt + T 3dT 3 dt + T 4dT 4 dt � = 0: Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 373 O.A. Goncharova By substituting the expressions for T i, one gets d2a2 dt2 � 2c1 da2 dt � 2c0k2a 3 � + d2a3 dt2 � 2c1 da3 dt + k1a 4 + 2c1k2a 2 � 2c1k3a 4 � + d2a4 dt2 � 2c1 da4 dt � k1a 3 + 2c0k3a 3 � + ::: = 0: (5) As a vector �eld a(t) is unit, then (a2)2+(a3)2+(a4)2 = 1. Di�erentiate this condition twice d2a2 dt2 (2a2) + d2a3 dt2 (2a3) + d2a4 dt2 (2a4) + ::: = 0: (6) The system of three di�erential equations (3), (5), (6) of the second order relatively to the coe�cients ai with additional conditions c(t)� c1d(t) = 0; (7) (a2)2 + (a3)2 + (a4)2 = 1; (8) a2 da2 dt + a3 da3 dt + a4 da4 dt = 0 (9) is obtained. If the determinant � of the matrix consists of the coe�cients of d 2 a i dt2 in (3), (5), (6) and is distinct from zero, then the system in its normal form can be written as follows: d2ak dt2 = Fk � t; ai; daj dt � ; k = 2; 3; 4: (10) Calculate the determinant of the matrix. In spite rather, of the expressions being rather complex, one gets a very simple result � = �k1 � a4 da3 dt � a3 da4 dt � k3((a 3)2 + (a4)2) + k2a 2a4 � � 2c1 � da2 dt � k2a 3 �2 �2c1 � da3 dt � k3a 4 + k2a 2 �2 �2c1 � da4 dt + k3a 3 �2 +2c1 � a2 da2 dt + a3 da3 dt + a4 da4 dt � = �k1(T 3a4 � T 4a3)� 2c1[(T 2)2 + (T 3)2 + (T 4)2]: Using the equation (4), one �nally obtains � = �c1[(T 2)2 + (T 3)2 + (T 4)2]: 374 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 Ruled Surfaces in E4 with Constant Ratio of the Gaussian Curvature... The system (10) of the second orders, when using a usual method, can be reduced to the system of equations of the �rst order. Let y be a vector with the components y2; : : : ; y7. Put y2 = a2, y3 = a4, y4 = a6, y5 = da 2 dt , y6 = da 3 dt , y7 = da 4 dt . Then the system can be written in the following form: dy dt = F (y; t): Choose the initial conditions of the system as follows: y20 = a2(t0) = 0; y30 = a3(t0) = 1; y40 = a4(t0) = 0; y60 = da3 dt (t0) = 0: Then additional conditions (7) and (8) are executed automatically. From (9) one gets the equation � da2 dt (t0)� k2 �2 = � da4 dt + k3 �� k1 c1 � � da4 dt + k3 �� : First, consider the case when the number c1 is positive. Let us assume y70 = da4 dt (t0) = �k3 + "; y50 = da2 dt (t0) = k2(t0) + s " � k1(t0) c1 � " � : It is possible to choose a positive small enough ", so that the additional condition (9) will take place. For c1 > 0 at point (t0; y0), the determinant is distinct from zero �(y0) = �c1 "� da2 dt (t0)� k2 �2 + � da4 dt (t0) + k3 �2 # = �c1 � " � k1 c1 � " � + "2 � = �"k1 6= 0: Now consider the case c1 < 0. Let us assume y70 = da4 dt (t0) = �k3 � "; y50 = da2 dt (t0) = k2(t0) + s �" � k1(t0) c1 + " � ; Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 375 O.A. Goncharova then it is possible to choose a positive small enough ", so that the additional condition (9) will take place. For c1 < 0 at point (t0; y0), the determinant is also distinct from zero �(t0) = �c1 � �" � k1 c1 + " � + "2 � = "k1 6= 0: Let us consider some neighborhood of point (t0; y0). The determinant � is a function of t; y2; y3; : : : ; y7: � = �(t; y2; : : : ; y7) which is the algebraic function of t; y2; : : : ; y7 and, consequently, is continuous. Thus, the existence of the ruled surfaces for which the ratio of the Gaussian torsion and curvature is constant follows from the existence theorem of the theory of ordinary di�erential equations. Consider the question of regularity of the constructed surfaces. The coe�- cients of their metric have the following form (see [2]): g11 = 1 + 2u(�1; a 0) + u2(a0)2; g12 = 0; g22 = 1: Then get the condition of regularity g11g22 = (1� uk1a 2) + u2((T 2)2 + (T 3)2 + (T 4)2) 6= 0: In point (t0; y0), the inequality 4P i=2 (T i)2 � "2 holds. From the above it follows that at all values of the parameter u in some neighborhood of point (t0; y0), the inequality g11g22 > 0 is valid. The surface is regular for the whole strip limited by extreme generatrix. Thus Theorem 1 is proved. Now let us turn to some examples of the ruled surfaces with a constant ratio of the Gaussian torsion and Gaussian curvature with some base curves. It is known that the tangent vector of a curve with constant ratio of torsion and curvature in E3 makes a constant angle with the �xed direction (see [3]). Denote these curves as slope curves in E3. Recall that at the beginning of the paper there was given the de�nition of slope curves in E4. For these curves k2 = �k1; k3 = �k1 (11) hold, where �; � are constants, and � 6= 0. Frene's equations for this type of the curves, can be written in the form: d�1 ds = k1�2; d�2 ds = k1(��1 + ��3); d�3 ds = k1(���2 + ��4); d�4 ds = �k1��3: 376 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 Ruled Surfaces in E4 with Constant Ratio of the Gaussian Curvature... Introduce a new parameter t = R k1ds on the slope curve in E4. By di�eren- tiating several times d�1 dt and using Frene's equations, get the di�erential equation on �1 with constant coe�cients d4�1 dt4 + (1 + �2 + �2) d2�1 dt2 + �2�1 = 0: From the theory of di�erential equations it follows that the common solution of the equation above is �1 = A1 cos �1t+B1 sin�1t+A2 cos �2t+B2 sin �2t; where �1, �2 are some functions from �; �, and A1; B1; A2; B2 are constant vectors satisfying condition j�1j = 1. By integrating �1, it is possible to �nd a position vector of the slope curve in E4. Now prove Prop. A. Let us construct a generalized standard ruled surface along the slope curve in E4. So, let a base curve be a slope curve in E4 with certain conditions on the curvatures k1 6= 0, k2 = �k1, k3 = �k1, and the generatrix be directed along �3, that is a 2 � 0, a3 � 1, a4 � 0. Then the equation b(t) = 0 is reduced to k02k3 � k2k 0 3 = 0 which is executed by virtue of (11). From (2) it follows that �k1k3 = �c1(k 2 2 + k23), that is c1 = k1k3 k 2 2 +k2 3 = � �2+�2 . It appears that the ratio of the Gaussian torsion and curvature at the gener- alized standard ruled surfaces along the slope curve in E4 is constant. From the expression g11g22 it is easy to see that the surface is regular. Proposition A is proved. If the curvatures of the curve are constant, then this case may be referred to the standard ruled surface �3 in E4 described in [4]. To prove Prop. B, construct an example of the ruled surfaces with a constant ratio of the Gaussian torsion and Gaussian curvature along a �at curve. In this case k2 = 0, k3 = 0, and the functions b(t), c(t), d(t) can be written in the following form: b(t) = � da4 dt d2a3 dt2 � da3 dt d2a4 dt � a2 + da2 dt � d2a4 dt2 a3 � d2a3 dt2 a4 � + � da3 dt a4 � da4 dt a3 �� d2a2 dt2 � k21a 2 � ; c(t) = k1 � da3 dt a4 � da4 dt a3 � ; (12) d(t) = � "� da2 dt �2 + � da3 dt �2 + � da4 dt �2 # : (13) Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 377 O.A. Goncharova If a2 � 0, then the equation b(t) = 0 is executed identically. We put the coe�cients a3 = sin'(t); a4 = cos'(t) (14) and substitute them into expressions for c(t) and d(t). Then the equation (2) by virtue of (12) - (14) has the following form: k1 d' dt + c1 � d' dt �2 = 0: In the case when d' dt = 0, the surface is a cylinder for which K = 0. In this paper the surfaces of this type are not considered. Hence, d' dt = � k1(t) c1 : The solution of the di�erential equation is as follows: '(t) = � 1 c1 tZ 0 k1(t)dt+ '0: As (T 3)2 + (T 4)2 = � d' dt �2 6= 0, then the surface is regular. Proposition B is proved. Choose a closed �at curve with the length `. If any point on this curve passes the whole curve, then the increment �' of the angle ' is equal to �'(t) = � 1 c1 R̀ 0 k1(t)dt. When c1 = 1 2� `Z 0 k1(t)dt; then �' = �2�, and in this case the vector a(t) will return to the initial position. Thus we obtain the surface that is homeomorphic to the cylinder with the base �at curve. A special case of the �at closed curve is a circle, i.e., k1 = const. If any point on the circle passes the whole circle, then the angle ' gets the increment �' = � k1 c1 `, where ` is the length of the circle, that is �' = �2� c1 . If c1 = p q is a rational number, and the point of the curve passes it p times, then �' is multiple 2�. In this case the vector a(t) will return to the initial position, and we obtain the surface that is homeomorphic to the cylinder with the circle taken p times as a base curve. If c1 is an irrational number, then the vector a(t) will not return to the initial position. Thus, if the point of the circle passes it in�nite number of times, we get the surface homeomorphic to a plane. 378 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 Ruled Surfaces in E4 with Constant Ratio of the Gaussian Curvature... References [1] Yu.A. Aminov, Surfaces in E4 with Gaussian Curvature Coinciding with Gaussian Torsion up to the Sign. � Math. Notes 56 (1994), No. 6, 1211�1215. [2] O.A. Goncharova, Ruled surfaces in En. � J. Math. Phys., Anal., Geom. 2 (2006), 40�61. (Russian) [3] Yu.A. Aminov, Di�erential Geometry and Topology of Curves. Gordon and Breach Sci. Publ., Amsterdam, 2000. [4] O.A. Goncharova, Standard Ruled Surfaces in En. � Dop. NAN Ukr. 3 (2006), 7�12. (Russian) Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 3 379

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