Conditions on a Surface F² is subset of Eⁿ to lie in E⁴

We consider a surface F² in Eⁿ with a non-degenerate ellipse of normal curvature whose plane passes through the corresponding surface point. The definition of three types of points is given in dependence of the position of the point relatively to the ellipse. If in the domain D is subset of F² all t...

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Date:	2013
Main Authors:	Aminov, Yu.A., Nasiedkina, Ia.
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Cite this:	Conditions on a Surface F² is subset of Eⁿ to lie in E⁴ / Yu.A. Aminov, Ia. Nasiedkina // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 2. — С. 127-149. — Бібліогр.: 13 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1067422025-02-23T19:07:43Z Conditions on a Surface F² is subset of Eⁿ to lie in E⁴ Aminov, Yu.A. Nasiedkina, Ia. We consider a surface F² in Eⁿ with a non-degenerate ellipse of normal curvature whose plane passes through the corresponding surface point. The definition of three types of points is given in dependence of the position of the point relatively to the ellipse. If in the domain D is subset of F² all the points are of the same type, then the domain D is said also to be of this type. This classification of points and domains is linked with the classification of partial differential equations of the second order. The theorems on the surface to lie in E⁴ are proved under the fulfilment of certain boundary conditions. Some examples of the surfaces are constructed to show that the boundary conditions of the theorems are essential. Рассмотрена поверхность F² в Eⁿ с невырожденным эллипсом нормальной кривизны, плоскость которого проходит через соответствующую точку поверхности. Дано определение трех типов точек на поверхности в зависимости от расположения точки относительно этого эллипса. Если в области D из F² все точки принадлежат одному типу, то говорим, что область D также принадлежит к этому типу. Эта классификация точек и областей оказывается связанной с классификацией дифференциальных уравнений в частных производных второго порядка. Доказаны теоремы о принадлежности поверхности к E⁴ при выполнении определенных краевых условий. Построены примеры поверхностей, показывающие, что краевые условия существенны. 2013 Article Conditions on a Surface F² is subset of Eⁿ to lie in E⁴ / Yu.A. Aminov, Ia. Nasiedkina // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 2. — С. 127-149. — Бібліогр.: 13 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106742 en Журнал математической физики, анализа, геометрии application/pdf 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	We consider a surface F² in Eⁿ with a non-degenerate ellipse of normal curvature whose plane passes through the corresponding surface point. The definition of three types of points is given in dependence of the position of the point relatively to the ellipse. If in the domain D is subset of F² all the points are of the same type, then the domain D is said also to be of this type. This classification of points and domains is linked with the classification of partial differential equations of the second order. The theorems on the surface to lie in E⁴ are proved under the fulfilment of certain boundary conditions. Some examples of the surfaces are constructed to show that the boundary conditions of the theorems are essential.
format	Article
author	Aminov, Yu.A. Nasiedkina, Ia.
spellingShingle	Aminov, Yu.A. Nasiedkina, Ia. Conditions on a Surface F² is subset of Eⁿ to lie in E⁴ Журнал математической физики, анализа, геометрии
author_facet	Aminov, Yu.A. Nasiedkina, Ia.
author_sort	Aminov, Yu.A.
title	Conditions on a Surface F² is subset of Eⁿ to lie in E⁴
title_short	Conditions on a Surface F² is subset of Eⁿ to lie in E⁴
title_full	Conditions on a Surface F² is subset of Eⁿ to lie in E⁴
title_fullStr	Conditions on a Surface F² is subset of Eⁿ to lie in E⁴
title_full_unstemmed	Conditions on a Surface F² is subset of Eⁿ to lie in E⁴
title_sort	conditions on a surface f² is subset of eⁿ to lie in e⁴
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2013
url	https://nasplib.isofts.kiev.ua/handle/123456789/106742
citation_txt	Conditions on a Surface F² is subset of Eⁿ to lie in E⁴ / Yu.A. Aminov, Ia. Nasiedkina // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 2. — С. 127-149. — Бібліогр.: 13 назв. — англ.
series	Журнал математической физики, анализа, геометрии
work_keys_str_mv	AT aminovyua conditionsonasurfacef2issubsetofentolieine4 AT nasiedkinaia conditionsonasurfacef2issubsetofentolieine4
first_indexed	2025-11-24T14:31:15Z
last_indexed	2025-11-24T14:31:15Z
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fulltext	Journal of Mathematical Physics, Analysis, Geometry 2013, vol. 9, No. 2, pp. 127–149 Conditions on a Surface F 2 ⊂ En to lie in E4 Yu.A. Aminov and Ia. Nasiedkina Mathematics Division, B. Verkin Institute for Low Temperature Physics and Engineering National Academy of Sciences of Ukraine 47 Lenin Ave., Kharkiv 61103, Ukraine E-mail: aminov@ilt.kharkov.ua Received May 30, 2011, revised January 30, 2012 We consider a surface F 2 in En with a non-degenerate ellipse of normal curvature whose plane passes through the corresponding surface point. The definition of three types of points is given in dependence of the position of the point relatively to the ellipse. If in the domain D ⊂ F 2 all the points are of the same type, then the domain D is said also to be of this type. This classification of points and domains is linked with the classification of partial differential equations of the second order. The theorems on the surface to lie in E4 are proved under the fulfilment of certain boundary conditions. Some examples of the surfaces are constructed to show that the boundary conditions of the theorems are essential. Key words: an ellipse of normal curvature, asymptotic lines, character- istics, boundary conditions. Mathematics Subject Classification 2000: 53A05. 1. Introduction Conditions for a two-dimensional surface in E4 to lie in a hyperplane E3 are well known [1, p. 146]. In this paper we consider the surface F 2 in En with non-degenerate ellipse of normal curvature such that the plane of this ellipse for each point x of the surface passes through this point x. This condition means that the point codimension of the surface is equal to 2. Notice that the condition on the plane is fulfilled for all surfaces lying in E4. We say that the point x ∈ F 2 is of 1) hyperbolic type if x lies outside the ellipse of normal curvature, 2) parabolic type if x lies on this ellipse, 3) elliptic type if x lies inside this ellipse. This paper is prepared with the support of grants from National Academy of Sciences of Ukraine and Russian Foundation for Basic Researches, 2012. c© Yu.A. Aminov and Ia. Nasiedkina, 2013 Yu.A. Aminov and Ia. Nasiedkina If all points of some domain G of a surface are of one of the types, then the domain G is said also to be of this type. If the domain G has the points of different types, then it is said to be of mixed type. Below we will connect this splitting into types with the classification of differential equations in partial derivatives of the second order. We will also give the conditions on the surfaces F 2 ⊂ En of types 1)–3), under which F 2 is also to lie in some E4. 2. About the Surfaces of Hyperbolic Type in En Let us consider the surface F 2 with the non-degenerate ellipse of normal curvature in the space En such that all points x ∈ F 2 are of hyperbolic type. We recall the definition of the normal curvature ellipse. Denote by kn(τ) the vector of normal curvature of F 2 determined by the tangent vector τ ∈ Tx at every point x ∈ F 2. The vector kn(τ) is in the normal space Nx. Let the start of this vector lie at the point x. Then the set of end-points of vectors kn(τ), when τ rotates in Tx, forms the indicatrix of normal curvature. This set is a closed plane curve, namely an ellipse, perhaps degenerated at a segment or a point. If τ is defined by the differentials of du1, du2, and Lα ijduiduj are the second fundamental forms with respect to the unit normals nα, α = 1, . . . , n− 2, then kn(τ) = Lα ijduiduj ds2 nα, where ds2 = gijduiduj is the metric form. In a normal space Nx, we introduce the Cartesian coordinates Xα with origin at the point x and basic vectors nα. Then the coordinates of the point M of the indicatrix have the form Xα = Lα ijduiduj gijduiduj . (1) On the surface F 2 we construct geometrically some net of curves. In this case, through the point x it is possible to draw two straight lines tangent to the ellipse. We denote the points of contacts by P and Q. Each of this points corresponds to the vector of normal curvature knP or knQ. In the tangent plane of F 2 for the vector knP there exists a tangent direction such that the vector of normal curvature for this direction coincides with knP . We have this direction at each point of the considering domain and hence we have the field of tangent directions. This field generates some family of integral curves which we call characteristics. With the help of the vector knQ we get another family of curves — also characteristics. The curves of these two families are not tangent to each other. On the surface, these families form some net of curves which we will call specially hyperbolic net. 128 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 Conditions on a Surface F 2 ⊂ En to lie in E4 Theorem 1. Let the surface F 2 ⊂ En of the class C4 with non-degenerate ellipse of normal curvature be of hyperbolic type. And let D be a triangular domain on F 2 bounded by two characteristics — curves η1 and η2 from different families of the specially hyperbolic net beginning at the point x, and some curve γ of the class C2 which crosses η1, η2 and is not tangent to the characteristics. Assume that there exists some hyperplane E4 such that both the curve γ ⊂ E4 and the tangent surface strip along γ lie in E4. Then the whole domain D lies in E4. P r o o f. On the surface of the domain D introduce some coordinates u1, u2 of the class C4. Then the fundamental forms of F 2 are the following: ds2 = gijduiduj , IIα = Lα ijduiduj . The normals to F 2 will be chosen as follows. Let the vectors n1, n2 define the plane of the ellipse. Other normals nk, k = 3, 4, . . . , n − 2, are taken to be orthogonal to this plane. Hence Lk ij = 0, k = 3, 4, . . . , n− 2. The Gauss decomposition has the form rij = Γk ijrk + Lα ijnα. Consider this decomposition for i = 1, j = 2 r12 = Γk 12rk + Lα 12nα. Due to the choice of normals, we have r12 = Γk 12rk + L1 12n1 + L2 12n2. (2) Lemma 1. The coefficients L̄1 12, and L̄2 12 in the special hyperbolic system of coordinates are equal to zero. P r o o f. We construct a new system of the coordinates ξ, η such that the coordinate lines are the characteristics. In this system, the coefficients of the second fundamental forms L̄α 12 are L α 12 = Lα ij ∂ui ∂ξ ∂uj ∂η , α = 1, 2. Take a straight line which does not cross the ellipse when passing through x. Then the normal n1 is the direction vector of this straight line, and n2 is a vector orthogonal to n1. We suppose that n2 is directed to the half-plane where the ellipse lies. Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 129 Yu.A. Aminov and Ia. Nasiedkina Let M be an arbitrary point outside the ellipse. Then its coordinate Y > 0. For the non-zero shift du1, du2, we have Y = L2 ijduiduj ds2 > 0. If ϕ is the angle between the strait line xM and n1, then ctgϕ = X Y = L1 ijduiduj L2 ijduiduj . The extremal value of the angle ϕ will be attained when the straight line xM is the tangent of the ellipse. If the ellipse is non-degenerate, then we have two extremal values of ϕ and two straight lines xM that are tangents of the ellipse. As in the case with the main directions of the 2-dimensional surface in E3, for determining the corresponding directions in the tangent plane of F 2, it is necessary to solve the equation \|L1 ij − λL2 ij \| = 0. Let λ1 and λ2 be the roots of this equation, and τ = {τ i}, ν = {νi} be two directions {du1, du2} in the tangent plane of F 2 corresponding to λ1 and λ2. Then the following system takes place: (L1 11 − λ1L 2 11)τ 1 + (L1 12 − λ1L 2 12)τ 2 = 0, (L1 11 − λ1L 2 12)τ 1 + (L1 22 − λ1L 2 22)τ 2 = 0, (L1 11 − λ2L 2 11)ν 1 + (L1 12 − λ2L 2 12)ν 2 = 0, (L1 12 − λ2L 2 12)ν 1 + (L1 22 − λ2L 2 22)ν 2 = 0. Multiply the first equation on ν1, the second equation on ν2, and take their sum to obtain L1 ijτ iνj − λ1L 2 ijτ iνj = 0. Similarly, for the third and fourth equations we have L1 ijτ iνj − λ2L 2 ijτ iνj = 0. Therefore, (λ1 − λ2)L2 ijτ iνj = 0. Because (λ1 − λ2) 6= 0, we have L2 ijτ iνj = 0 and L1 ijτ iνj = 0. We take the functions η(u1, u2) and ξ(u1, u2) such that the lines η = const are the integral 130 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 Conditions on a Surface F 2 ⊂ En to lie in E4 curves of the vector field τ , and the lines ξ = const are the integral curves of the field ν. Then, ηu1 = −qτ2, ξu1 = −pν2, ηu2 = qτ1, ξu2 = pν1, with some functions p 6= 0 and q 6= 0. Let us write the transformation from the coordinates ξ, η to u1, u2 u1 = u1(ξ, η), u2 = u2(ξ, η). If J = J ( u1,u2 ξ,η ) is the Jacobian of the transformation, then ξu1 = 1 J ∂u2 ∂η , ηu1 = − 1 J ∂u2 ∂ξ , τ1 = 1 qJ ∂u1 ∂ξ , ν1 = − 1 pJ ∂u1 ∂η , ξu2 = − 1 J ∂u1 ∂η , ηu2 = 1 J ∂u1 ∂ξ , τ2 = 1 qJ ∂u2 ∂ξ , ν2 = − 1 pJ ∂u2 ∂η . Hence, 0 = Lα ijτ iνj = − 1 pqJ2 Lα ij ∂ui ∂ξ ∂uj ∂η = − 1 pqJ2 L̄α 12. Therefore, L̄α 12 = 0. Lemma 1 is proved. Lemma 2. The surface F 2 with respect to the coordinates ξ, η belongs to the regularity class C2. P r o o f. For the proof of this lemma we will need the result of the following lemma. Lemma 3. Suppose F 2 has the regularity of the class C4 with respect to the coordinates (u1, u2). Then the coefficients Lα ij = (rij , nα) have the regularity of the class C2. P r o o f. First, we will find the regularity class of the normals nα, α = 3, . . . , n − 2 . As the plane of the ellipse passes through x, the vectors rij lie in the 4-dimensional space spanned by ru1 , ru2 , n1, n2. Therefore, five vectors ru1 , ru2 , ru1u1 , ru1u2 , ru2u2 are linearly dependent. As the ellipse of normal curva- ture is non-degenerate, there exist four linearly independent vectors between the first and the second derivatives of r, for example, ru1 , ru2 , ru1u2 , ru2u2 . Let the normal nα have the coordinates ξj , j = 1, . . . , n. To determine the normals, we can write the following system: Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 131 Yu.A. Aminov and Ia. Nasiedkina (ru1 , nα) = x1u1 ξ1 + x2u1 ξ2 + . . . + xnu1 ξn = 0, (ru2 , nα) = x1u2 ξ1 + x2u2 ξ2 + . . . + xnu2 ξn = 0, (ru1u2 , nα) = x1u1u2 ξ1 + x2u1u2 ξ2 + . . . + xnu1u2 ξn = 0, (ru2u2 , nα) = x1u2u2 ξ1 + x2u2u2 ξ2 + . . . + xnu2u2 ξn = 0. The equations of this system are linearly independent. Let, for example, the determinant be not equal to zero ∆ = ∣∣∣∣∣∣∣∣ x1u1 . . . x4u1 x1u2 . . . x4u2 x1u1u2 . . . x4u1u2 x1u2u2 . . . x4u1u2 ∣∣∣∣∣∣∣∣ 6= 0. By solving the system of the linear equations, we get ξ1 = ∆−1(ξ5A1 + ξ6A2 + . . . + ξnAn−4), ξ2 = ∆−1(ξ5B1 + ξ6B2 + . . . + ξnBn−4), ξ3 = ∆−1(ξ5C1 + ξ6C2 + . . . + ξnCn−4), ξ4 = ∆−1(ξ5D1 + ξ6D2 + . . . + ξnDn−4), where ∆, Ai, Bi, Ci, Di are some minors of the regularity of the class C2. Setting the values ξ5, . . . , ξn, we get the n − 4 normals. Hence, the normals nα, α = 3, . . . , n − 2 have the regularity of the class C2. To obtain the regular fields of the normals n1, n2, we write the system (ruk , ni) = 0, i = 1, 2, (nα, ni) = 0, α = 3, . . . , n− 2. As in the above, we get that ni, i = 1, 2 belong to the class C2. Hence, Lα ij ∈ C2. Remark that in the given construction the vectors nj may not be orthogonal to each other. But it is not difficult to check that the process of orthogonality gives new normals of the same class of regularity. Lemma 3 is proved. Continue the proof of Lemma 2. From the equation \|L1 ij − λL2 ij \| = 0 and Lemma 3 it follows that λi ∈ C2. Therefore, the fields τ and ν are of the class C2. We have the system of differential equations of the first order ξu1ν 1 + ξu2ν 2 = 0, ηu1τ 1 + ηu2τ 2 = 0. 132 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 Conditions on a Surface F 2 ⊂ En to lie in E4 The coefficients of this system are of the class C2. Then, by the theorem on the regular dependence of solutions of an ordinary differential equation on initial data (see, for example, [10, p. 92 § 2] and for the existence of the solution see [10, p. 255], the functions ξ, η also belong to the class C2. Lemma 2 is proved. Equation (2) can be written in the canonical form rξη = Γ1 12rξ + Γ2 12rη. (3) Both the considering curve γ and the surface strip lie in some space E4. Let n0 be a constant vector from a normal space orthogonal to E4. Denote U = (r, n0). From (3) it follows that Uξη = Γ1 12Uξ + Γ2 12Uη. (4) We suppose that the origin of the Cartesian coordinate in En lies in E4. The conditions of the theorem imply that U \|γ = 0, Uν \|γ = 0, where Uν is the derivative on F 2 at the direction orthogonal to γ. According to the theorem on the uniqueness of solutions in the theory of differential equations of the second order of hyperbolic type (see [2, p. 439], [3, p. 65]), we may conclude that U(ξ, η) ≡ 0 in the domain D. It means that D lies in the hyperplane orthogonal to n0. If we take n0 as an arbitrary constant vector in the space orthogonal to E4, then the domain D lies in E4. 3. About the Surfaces of Parabolic Type in En Consider the case when every point x ∈ F 2 lies on the ellipse of normal curvature corresponding to this point. In the plane Tx, which is tangent to the surface F 2, there exists the direction τ for which kn(τ) = 0. The integral curves of the field of the vectors τ are asymptotic curves in the usual sense of differential geometry. Introduce a system of the coordinates (ξ, η) on the surface with a family of asymptotic lines as ξ-lines. The second family of the coordinate η-lines can be taken arbitrarily as a regular family of curves crossing asymptotic lines transver- sally. For example, we can take the family of orthogonal trajectories of the family of asymptotic lines. Then the vector position r of F 2 is r = r(ξ, η). Write the Gauss decomposition with i = j = 1, Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 133 Yu.A. Aminov and Ia. Nasiedkina r11 = Γk 11rk + Lα 11nα. As the ξ-lines are asymptotic, Lα 11 = 0, α = 1, . . . , n− 2. For r(ξ, η), we have the parabolic equation r11 = Γk 11rk. Theorem 2. Let the analytical surface F 2 ⊂ En with the ellipse of normal curvature of non-degenerate type be of parabolic type. Let the domain D ⊂ F 2 be some strip bounded by two asymptotic curves. Suppose that some curve γ, crossing transversally all asymptotic lines in the strip and the tangent surface strip along γ lie in the some subspace E4. Then D also lies in this subspace E4. P r o o f. For proving, we introduce the function U = (r, n0), where n0 is the normal vector from the space orthogonal to E4. We get the equation Uξξ = Γ1 11Uξ + Γ2 11Uη. (5) Suppose that γ is given by the equation ξ = 0. Conditions on the curve γ give zero conditions for the Cauchy problem U(0, η) = 0, Uξ(0, η) = 0. By the Cauchy–Kovalevskaya theorem [11, p. 22], in the neighborhood of γ (that is, ξ = 0) there exists only one solution U ≡ 0. Hence, by the analyticity of F 2, the domain D lies in the subspace orthogonal to n0. By the arbitrariness of n0, D lies in E4. In the theory of parabolic differential equations there are other uniqueness theorems. We can formulate and prove the theorem not supposing analyticity of the surface, but imposing additional conditions. Theorem 3. Let the surface F 2 ⊂ En of the regularity class C5 with non- degenerate ellipse of normal curvature be of parabolic type. Let the domain D ⊂ F 2 be bounded by two asymptotic lines and by two curves γ and γ1 crossing the asymptotic lines transversally. Suppose that the geodesic curvature of asymptotic lines in D is nonnegative. Let the curve γ lie in some E4 together with the tangent surface strip along γ. Then the whole domain D lies in E4. 134 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 Conditions on a Surface F 2 ⊂ En to lie in E4 P r o o f. For the proof of this theorem, we use the theorem on the uniqueness of solutions for differential equations of parabolic type proved by E.M. Landis [4], where the equation ∂2U ∂x2 = a(t, x) ∂U ∂t + b(t, x) ∂U ∂x + c(t, x)U (6) is considered in some domain G. On the coefficients of this equation the following conditions are imposed: 1. The modules of the coefficients are limited by 1, 2. The coefficient a(t, x) has a derivative with respect to t, the coefficient b(t, x) has a derivative with respect to x, and ∣∣∣∣ ∂a ∂t ∣∣∣∣ < 1, ∣∣∣∣ ∂b ∂x ∣∣∣∣ < 1. 3. In the domain G a ≥ 0, c ≥ 0. 4. The solution U(x, t) of (6) in G is of the class C2. In [4], the following uniqueness theorem is proved. Theorem 4. Let G be a part of the strip Π = t1 < t < t2 situated between two non-crossing curves Γ1 and Γ2 with one-to-one projections on the t-axis and connecting the opposite sides of Π. Let U(x, t) be a solution of (6) belonging to C1 in D̄. Suppose that Γ2 is a smooth curve, and U \|Γ2= ∂U ∂n \|Γ2= 0. Then in G U ≡ 0. Here ∂U ∂n is a derivative in the direction orthogonal to Γ2. Give an explanation of the nonnegative geodesic curvature of asymptotic line. If the coordinates ξ, η are introduced in D, then on the basic curve η = 0 the vector of the curvature is directed inside D. The curvature vector of asymptotic lines inside D builds an acute angle or π 2 with positive direction of the η-line. To apply the Landis theorem, we have to verify the conditions on the coef- ficients. We put x = ξ and t = η. Then the coefficients from Condition 3 in Theorem 4 are a(t, x) = Γ2 11, b(t, x) = Γ1 11, and c(t, x) = 0 in our case. Remark that Γk ij ∈ C2 and, hence, in the domain D both the functions and the derivatives are limited. The boundedness condition of modules by 1 can be satisfied by a new parametrization of the coordinate lines. Remark that Γ2 11 is connected with the geodesic curvature of asymptotic line. Indeed, Γ2 11 = 1 2W 2 (2g11 ∂g12 ∂ξ − g12 ∂g11 ∂ξ − g11 ∂g11 ∂η ), Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 135 Yu.A. Aminov and Ia. Nasiedkina where gij are the coefficients of the metric, and W = √ g11g22 − g2 12. The geodesic curvature 1 ρg can be written as (see [5, § 83, 127] 1 ρg = 1 2Wg 3 2 11 (2g11 ∂g12 ∂ξ − g12 ∂g11 ∂ξ − g11 ∂g11 ∂η ). Comparing these two expressions, we have Γ2 11 = g 3 2 11 W 1 ρg . (7) Hence, the sign Γ2 11 depends on the sign of 1 ρg . By Condition 3 in Theorem 3, the coefficient a(x, t) ≥ 0 is satisfied if 1 ρg ≥ 0. The boundary conditions on γ will coincide with the boundary conditions of Landis’ theorem [4] if we take γ instead of Γ2. Thus, in D we have U(ξ, η) ≡ 0. Therefore, D ⊂ E4. In the theory of parabolic equations, differential equations are usually written in the form of the generalized heat equations, namely with the derivative chosen with respect to t (see [6]). This theorem will be used for proving the theorem bellow. Determine a non-closed contour Ω. Suppose the point C has the coordinates (a, T ); the point A has the coordinates (a, 0); the point B has the coordinates (b, 0) and the point K has the coordinates (b, T ) (here we suppose that a < b, T > 0). Then the contour Ω consists of the intercepts of coordinate lines. Consider the domain D with the boundary CABKC. Theorem 5. Suppose that F 2 ⊂ En of the class C5 with non-degenerate ellipse of normal curvature is of parabolic type. Let the contour Ω be situated in some space of E4. Let the geodesic curvature of asymptotic lines in D be positive. Then the whole domain D lies in E4. P r o o f. Begin with considering the contour Ω. The intercept AB is an asymptotic line of F 2 and, consequently, the tangent surface strip along AB lies in E4 automatically. Although, on CA and BK the condition on the tangent surface strip is absent. The contour Ω consists of an intercept of the coordinate line η = 0 and two intercepts of the coordinate η-lines. Rewrite Eq. (5) in the form of the generalized heat equation Uη = 1 Γ2 11 Uξξ + Γ1 11 Γ2 11 Uξ. (8) By the condition 1 ρg > 0 and (7), we have Γ2 11 6= 0. Hence the coefficients of this equation have the regularity of the class C2. For (8), we apply the uniqueness 136 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 Conditions on a Surface F 2 ⊂ En to lie in E4 theorem for parabolic equations (see [6]) and obtain that U ≡ 0 in D. The Theorem 5 is proved. Consider now an infinitely long strip bounded by two complete asymptotic lines. We suppose that a system of the coordinates (ξ, η) in this strip can be introduced by using the family of asymptotic lines and the second family of the lines crossing the curves of the first family transversally. For example, it is possible to construct an orthogonal coordinates system where the strip is bounded by the basic curve Γ : η = 0 and by the curve Γ1: η = T which is said to be free. Let the parameter ξ be the arc length on Γ. Define the width of the strip. Every point P ⊂ Γ1 has some coordinates (ξ, T ). Let l(ξ) be the shortest distance between P and Γ along F 2. The number l(ξ) is called the variable width of the strip. Theorem 6. Suppose that F 2 ⊂ En of the class C5 with non-degenerate ellipse of normal curvature is of parabolic type. Suppose that D is an infinite strip between two complete asymptotic lines Γ and Γ1. Suppose that the asymptotic lines in D have a positive geodesic curvature, and the following conditions in D are fulfilled: ∣∣∣∣ 1 Γ2 11 ∣∣∣∣ ≤ M, ∣∣∣∣ Γ1 11 Γ2 11 ∣∣∣∣ ≤ M(\|ξ\|+ 1), \|l(ξ)\| ≤ Beβξ2 , B, β > 0, where M, B, and β are some positive numbers. If the basic curve Γ lies in some E4, then the whole domain D lies in E4. P r o o f. We apply Theorem 7 from [8, p. 63], which is a generalization of A.N. Tikhonov’s theorem from [9] proved for the classical heat equation. Let x = (x1, . . . , xn) be a varying point in En. Suppose that in the domain D, the function U(x, t) has continuous second derivatives with respect to x and continuous first derivatives with respect to the parameter t. The theorem is formulated as follows. Theorem 7. Let L be a parabolic operator L(U) = n∑ i,j=1 aij(x, t) ∂2U ∂ui∂uj + n∑ i,j=1 bi(t, x) ∂U ∂xi + c(x, t)U − ∂U ∂t , where ∑ aij(x, t)ξiξj > 0 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 137 Yu.A. Aminov and Ia. Nasiedkina for all vectors ξ = {ξi} 6= 0 with continuous coefficients in Rn × (0, T ], and the following conditions be satisfied: \|aij(x, t)\| ≤ M, \|bi(x, t)\| ≤ M(\|x\|+ 1), \|c(x, t)\| ≤ M(\|x\|2 + 1). Then there exists not more than one solution of the equation L(U) = f(x, t) in Rn × (0, T ] such that U(x, 0) = ϕ(x) in Rn and whenever \|U(x, t)\| ≤ Bexpβ\|x\|2, B, β > 0, (9) where M,B and β are some positive constants. Remark that for the heat equation (9), the condition from [9] is essential. Namely, if \|U(x, t)\| ≤ Beβ(x2+ε) and ε > 0, then the uniqueness theorem is not true. Apply this theorem to Eq. (8) under the conditions f(x, t) = 0, φ(x) = 0. Remark that \|U(ξ, η)\| is equal to a distance from the corresponding point of the surface to E4. By the condition of Theorem 6, we have \|U(ξ, η)\| ≤ l(ξ) ≤ Beβξ2 . Consequently, all conditions of Theorem 7 are fulfilled. Hence, under the given boundary conditions and limitations imposed on the coefficients, Eq. (8) has the unique solution U(ξ, η) ≡ 0. Thus, D lies in E4. Finally, Theorem 6 is proved. Give a few examples of the surface strips whose metrics satisfy the conditions of Theorem 6: 1) A universal covering of a ring on a plane between two concentric circles. The exterior contour is taken as a basic curve. The metric of the plane can be written in the form ds2 = η2dξ2 + dη2. Then Γ1 11 = 0, Γ2 11 = −η, and Eq. (8) takes the form uη = − 1 ηuξξ. 2) A ring on a half-sphere. 3) A strip on the Lobachevsky plane to be considered below in Sec. 5. 4) A local convex infinitely long curve γ with the radius position ρ(ξ) on the plane. The curvature k of γ satisfies the restrictions 0 < k1 ≤ k ≤ k2, where k1, k2 are constants. The strip consists of the family of parallel curves. The vector position of a point in the plane has the form r(ξ, η) = ρ(ξ) + ην(ξ), where ν(ξ) 138 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 Conditions on a Surface F 2 ⊂ En to lie in E4 is a unit normal to γ. The metric form of the plane is ds2 = (1− ηk)2dξ2 + dη2. Introduce the restrictions on η: 0 < η < 1−ε k2 , where 0 < ε < 1 and \|kξ\| ≤ M = const. Then, ∣∣∣∣ 1 Γ2 11 ∣∣∣∣ ≤ 1 εk1 , ∣∣∣∣ Γ1 11 Γ2 11 ∣∣∣∣ ≤ ∣∣∣∣ (1− ε)kξ k2ε2k1 ∣∣∣∣ ≤ M1 = const. Thus all conditions of Theorem 6 for this strip are fulfilled. 4. On the Surfaces of Elliptic Type in En Theorem 8. Let the domain D be homeomorphic to a disk and be of elliptic type. Assume that the surface has the regularity of the class C4,α. Suppose the boundary of D is a curve γ ∈ C1 which lies in some E4. Then the whole domain D lies in E4. P r o o f. In contrast to the theorems from Sections 1–3, the condition that the tangent surface strip along γ lies in E4 is not necessary. Theorem 8 can be proved by using the theory of elliptic equations or some geometrical considerations. First we are to obtain an elliptic equation for the vector position r(x1, x2) of F 2 with the coordinates x1, x2. Suppose that the coordinates x1, x2 are intro- duced in the whole simply connected domain D. Write the Gauss expansions by using the covariant derivatives r,ij = Lα ijnα. Recall that Lα ij ≡ 0, α = 3, . . . , n−2 because the plane of the ellipse of normal curvature passes through x ∈ F 2. Multiply the right- and the left-sides of the Gauss equations by some numbers Ωij r,ij Ωji = Lα ijΩ jinα. Assume that Ωji have the following properties: Lα ijΩ ji = 0, α = 1, 2 and Ωij = Ωji. Then we obtain the equation for the vector position r(x1, x2) r,11 Ω11 + 2r,12 Ω12 + r,22 Ω22 = 0. (10) Write Ωij in terms of the coefficients Lα ij Ω11 = ∣∣∣∣ L1 12 L1 22 L2 12 L2 22 ∣∣∣∣ , Ω12 = −1 2 ∣∣∣∣ L1 11 L1 22 L2 11 L2 22 ∣∣∣∣ , Ω22 = ∣∣∣∣ L1 11 L1 12 L2 11 L2 12 ∣∣∣∣ . For Eq. (10) to be elliptic, the inequality (Ω12)2 − Ω11Ω22 < 0, should be fulfilled. Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 139 Yu.A. Aminov and Ia. Nasiedkina Determine the sign of (Ω12)2 − Ω11Ω22 which depends on disposition of the point x relatively to the ellipse. We use a special system of coordinates on F 2 and obtain the law of transformation for (Ω12)2 − Ω11Ω22 under the transition from x1, x2 to the coordinates u1, u2. Then, by the law of transformation of Ωji to Ωij , determine Ωij = J3Ωαβ ∂ui ∂xα ∂uj ∂xβ , where J(x1,x2 u1,u2 ) is a Jacobian of transformation. The obtained Ωij are said to be relative tensors with weight 3 (see [13, p. 237]). Consider, for example, the expressions Ω11 = ∣∣∣∣∣ L 1 12 L 1 22 L 2 12 L 2 22 ∣∣∣∣∣ , where L σ ij are the coefficients of the second quadratic form in a new coordinate system. The coordinates transformation influences on the coefficients of the sec- ond quadratic form in the following way: L σ ij = Lσ αβ ∂xα ∂ui ∂xβ ∂uj . Consequently, Ω11 = ∣∣∣∣∣ L1 αβ ∂xα ∂u1 ∂xβ ∂u2 L1 γδ ∂xγ ∂u2 ∂xδ ∂u2 L2 αβ ∂xα ∂u1 ∂xβ ∂u2 L2 γδ ∂xγ ∂u2 ∂xδ ∂u2 ∣∣∣∣∣ = J {∣∣∣∣ L1 11 L1 21 L2 11 L2 21 ∣∣∣∣ ( ∂x1 ∂u2 )2 + ∣∣∣∣ L1 11 L1 22 L2 11 L2 22 ∣∣∣∣ ∂x1 ∂u2 ∂x2 ∂u2 + ∣∣∣∣ L1 12 L1 22 L2 12 L2 22 ∣∣∣∣ ( ∂x2 ∂u2 )2 } . Substitute the determinants consisting of Lσ ij by Ωij and use the derivatives ∂uβ ∂xj instead of ∂xi ∂uα . We obtain Ω11 = J3 ( Ω22 ( ∂u1 ∂x2 )2 + 2Ω12 ∂u1 ∂x2 ∂u1 ∂x1 + Ω11 ( ∂u1 ∂x1 )2 ) = J3Ωαβ ∂u1 ∂xα ∂u1 ∂xβ . For the ellipticity to be preserved, the expression (Ω12)2−Ω11Ω22 should not change its sign when transforming from one coordinate system to another ∣∣∣∣∣ Ω11 Ω12 Ω12 Ω22 ∣∣∣∣∣ = J6 ∣∣∣∣∣ Ωαβ ∂u1 ∂xα ∂u1 ∂xβ Ωγδ ∂u1 ∂xγ ∂u2 ∂xδ Ωαβ ∂u2 ∂xα ∂u1 ∂xβ Ωγδ ∂u2 ∂xγ ∂u2 ∂xδ ∣∣∣∣∣ 140 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 Conditions on a Surface F 2 ⊂ En to lie in E4 = J5 ∂u1 ∂xβ ∂u2 ∂xδ ( Ω1βΩ2δ − Ω2βΩ1δ ) = J4 ( Ω11Ω22 − (Ω12)2 ) . (11) At it is seen from the last relation, the sign of (11) is not changed. It is easy to ascertain that under the rotation of normal basis n1, n2, the sign of the numbers Ωij is not changed. On the surface, take some point x0. In the neighborhood of x0, construct special orthogonal coordinate system in the following way. If at the point x0 the ellipse of normal curvature is not a circle, then take a vector of normal curvature whose end-point lies at one of summits of the ellipse (for example, at the largest one). Let the vector τ in the tangent plane be corresponding to the vector of normal curvature. Draw the coordinate u1-curve on F 2 tangential to τ at x0 and thus get the family of the u1-lines, namely the first family. If the ellipse of normal curvature is a circle, then τ can be taken arbitrarily. Taking the orthogonal trajectories of the first family, we get the second family of the coordinate lines. Chose the normals n1, n2 at x0 to be parallel to the axes of the ellipse (if it is not a circle), and take them arbitrarily in opposite case. Additionally, put g11 = g22 = 1 at x0. Under this choice of the coordinates and normals, at x0 we have L1 11 = α + a, L2 11 = β, L1 12 = 0, L2 12 = b, L1 22 = α− a, L2 22 = β, where α and β are the coordinates of the origin of the ellipse, and a, b are its half-axes. Consequently, at x0 we get (Ω12)2 − Ω11Ω22 = (∣∣∣∣ α + a α− a β β ∣∣∣∣ 2 − 4 ∣∣∣∣ 0 α− a b β ∣∣∣∣ ∣∣∣∣ α + a 0 β b ∣∣∣∣ ) = 4(β2a2 − b2a2 + α2b2) < 0. Geometrically, this inequality means that x0 lies inside the ellipse. If we introduce the function F (X1, X2) = (X1 − α)2 a2 + (X2 − β)2 b2 − 1 on the plane of the ellipse, then F = 0 for the points of this ellipse and F < 0 for the point x0 which has the coordinates X1 = 0,X2 = 0. Hence α2 a2 + β2 b2 − 1 < 0. Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 141 Yu.A. Aminov and Ia. Nasiedkina From Eq. (11), we obtain the elliptic equation for U = (r, n0), U11Ω11 + 2U12Ω12 + U22Ω22 = 0. Subsequently, from the uniqueness theorem for elliptic differential equations (see [2, p. 334], [12, p. 109]) there follows Theorem 8. By using another way of proving Theorem 8 proposed by A.A. Borisenko, an (n − 1)-dimensional sphere Sn−1 with the center in E4 containing the domain D is considered. This is a generalization of the method first suggested by A.V. Pogorelov. Introduce the Cartesian coordinates in En y1, . . . , yn such that the axes y1, . . . , y4 lie in E4, and y5, . . . , yn are orthogonal to E4. Consider the family of ellipsoids, obtained from Sn−1 by being compressed along the axes y5, . . . , yn to E4. If the domain D does not lie in E4, then at some moment of compression one ellipsoid touches D at some inner point x. The vectors of normal curvature at this point for every tangent vector are directed inside the ellipsoid. Consequently, the point x lies outside the ellipse of normal curvature. It means that x is of hyperbolic type which contradicts the theorem conditions. 5. Examples of Surfaces in E5 with the Plane of Normal Curvature Ellipse Passing through a Point of a Surface We show the way of constructing the surfaces in E5 with non-degenerate ellipse of normal curvature whose plane passes through the point x. The surfaces are constructed with a sufficiently large arbitrariness. The following theorem shows that the boundary conditions imposed on the previous theorems are essential. Theorem 9. Let Γ ⊂ E5 be a curve of the regularity class C5 with the curva- tures ki 6= 0, i = 1, . . . , 4. Then through Γ it is possible to draw a surface of the regularity class C2 whose ellipse of normal curvature is non-degenerate and the plane passes through the point x of this surface. P r o o f. If k4 6= 0, then neither the curve Γ nor the surface containing Γ lies in E4. The plane of the ellipse of normal curvature of every ruled surface in E5 passes through the point x. Moreover, every ruled surface with non-degenerate ellipse of normal curvature is of parabolic type. Our construction provides non- degeneracy of this ellipse. If the vector position of the curve Γ is ρ(s), where s is the arc length, then in E5 we take the vector position of the ruled surface r(s, t) = ρ(s) + tξ3(s). Here ξi are the vectors of natural basis of Γ. 142 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 Conditions on a Surface F 2 ⊂ En to lie in E4 Calculate the first and second fundamental forms for this surface. We have rs = ξ1 + t(−k2ξ2 + ξ4k3), rt = ξ3. From here it is seen that g11 = 1 + t2(k2 2 + k2 3), g12 = 0, g22 = 1. Further, rss = ξ1tk1k2 + ξ2(k1 − tk′2) + ξ3(−tk2 2 − tk2 3) + ξ4tk ′ 3 + ξ5k3k4t, rst = −ξ2k2 + ξ4k3, rtt = 0. Write the normals to the surface n1 = λ[ −k2ξ2 + k3ξ4 k2 2 + k2 3 − tξ1], λ = √ k2 2 + k2 3 1 + t2(k2 2 + k2 3) , n2 = k3ξ2 + k2ξ4√ k2 3 + k2 2 , n3 = ξ5. Calculate the coefficients of the second quadratic forms of the surface L1 11 = (n1, rss) = λ(−t2k1k2 − k1k2 − tk′2k2 − tk3k ′ 3 k2 3 + k2 2 ), L1 12 = λ, Lα 22 = 0, α = 1, 2, 3, L2 11 = (k1 − k′2t)k3 + k2k ′ 3t√ k2 3 + k2 2 , L2 12 = 0, L3 11 = k4k3t, L3 12 = 0. Check whether the normal curvature ellipse at the points of the curve Γ is non-degenerate. The coordinates of the ellipse are given by equation (1). On the surface F 2, consider the first fundamental form dl2. At t = 0, we have dl2 = (ds)2 + (dt)2. Then for the ellipse of normal curvature we get cosϕ = ds√ (ds)2 + (dt)2 , sinϕ = dt√ (ds)2 + (dt)2 . Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 143 Yu.A. Aminov and Ia. Nasiedkina Here the angle ϕ is formed by the direction τ and the coordinate line s. For the coordinates of indicatrix we can write Xα(ϕ) = Lα 11cos 2ϕ + 2Lα 12sinϕ cosϕ + Lα 22sin 2ϕ. After transformation of this expression, we have Xα(ϕ) = Lα 11 + Lα 22 2 + Lα 11 − Lα 22 2 cos 2ϕ + Lα 12sin 2ϕ. Recall that Lα 22 = 0. Let us introduce the vectors M =   L1 11 L2 11 L3 11   , N =   L1 12 L2 12 L3 12   . For the normal curvature ellipse at the points of the curve Γ not to degenerate into a segment of a straight line, it is necessary and sufficient for the vectors M and N be non-collinear. Hence, the minor, composed of the components of these vectors, is not equal to zero ∣∣∣∣ L1 11 L1 12 L2 11 L2 12 ∣∣∣∣ = ∣∣∣∣ L1 11 L1 12 L2 11 0 ∣∣∣∣ = k1k3 6= 0. On a continuity, the ellipse of normal curvature will also be non-degenerate in some neighborhood of the curve Γ. Theorem 9 is proved. Give another example of the surface of parabolic type. Suppose that the metric of surface is the metric of the Lobachevsky plane in the Poincaré interpre- tation ds2 = dξ2 + dη2 η2 . Suppose also that asymptotic lines are horocycles η = const. For this metrics we have Γ1 11 = 0, Γ2 11 = 1 η . Then Eq. (5) has the form rη = ηrξξ. If replacing t = η2 2 , ξ = x, then this equation can be written in a classical form of the classical heat conduction equation rxx = rt. (12) Thus, the classical object, namely the Lobachevsky plane, leads to the classical heat conduction equation. 144 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 Conditions on a Surface F 2 ⊂ En to lie in E4 Let some continuous limited curve in E5 with the vector position ρ = ρ(x), −∞ < x < +∞ be given. Applying the Poisson formula (see [6, p. 225], or [7, p. 481]), we can define the surface F 2 ⊂ E5 r(x, t) = 1 2 √ π +∞∫ −∞ 1√ t e− (x−ξ)2 4t ρ(ξ)dξ, where r(x, 0) = ρ(x). Thus, we can state that except the ruled surfaces there are also other surfaces of parabolic type whose vector positions satisfy heat equation (12). Therefore, Lα 11 = 0. Hence, a curve t = const is an asymptotic line on the surface. It means that the surface is of parabolic type. 6. About the Surfaces in E5 in the Explicit Form In this section, we consider the surface in E5 given in the explicit form. The plane of the ellipse of normal curvature for the point x is to pass through this point. Let e1, . . . , e5 be the fixed basis in E5 with the Cartesian coordinates x1, x2, u, v, w. Then the surface F 2 ⊂ E5 is given as follow: u = u(x1, x2), v = v(x1, x2), w = w(x1, x2). Write the vector position and its derivatives in the form r =   x1 x2 u(x1, x2) v(x1, x2) w(x1, x2)   , rx1 =   1 0 u1 v1 w1   , rx2 =   0 1 u2 v2 w2   , rxixj =   0 0 uij vij wij   . Write the normals n1, n2, n3 of F 2 in terms of the coordinates n1 = {ξi}, n2 = {ηi}, n3 = {ζi}, i = 1, . . . , 5. The system of equations for the normal n1 is the following: ξα + uαξ3 + vαξ4 + wαξ5 = 0, (13) where α = 1, 2. The similar systems also exist for n2 and n3. Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 145 Yu.A. Aminov and Ia. Nasiedkina It is obvious that L1 ij = ξ3uij + ξ4vij + ξ5wij , L2 ij = η3uij + η4vij + η5wij , (14) L3 ij = ζ3uij + ζ4vij + ζ5wij are the coefficients of the second fundamental forms. Lemma 4. If the plane of the ellipse of normal curvature is defined and it passes through the point x of the surface, then ∆ = ∣∣∣∣∣∣ L1 11 L2 11 L3 11 L1 12 L2 12 L3 12 L1 22 L2 22 L3 22 ∣∣∣∣∣∣ = 0. On contrary, if ∆ = 0, then the plane of this ellipse passes through the point x. P r o o f. The vector of normal curvature can be rewritten as kn = (L1 11n1 + L2 11n2 + L3 11n3) (du1)2 ds2 + 2(L1 12n1 + L2 12n2 + L3 12n3) du1du2 ds2 +(L1 22n1 + L2 22n2 + L3 22n3) (du2)2 ds2 . The expressions in brackets, i.e., Nkl = ∑ α Lα klnα k, l = 1, 2, are vectors. It is known that the end of the vector kn describes a flat curve, namely, an ellipse. As the plane of normal curvature ellipse passes through the point x of the surface, the vectors Nkl are coplanar. Hence we have ∆ = ∣∣∣∣∣∣ L1 11 L2 11 L3 11 L1 12 L2 12 L3 12 L1 22 L2 22 L3 22 ∣∣∣∣∣∣ = 0. (15) On contrary, if ∆ = 0, then three vectors Nkl are linearly dependent. There- fore, kn for all du1, du2 lies in one plane passing through the point x. The lemma is proved. Lemma 5. For a surface in the explicit form, the equation ∆ = 0 gives ∣∣∣∣∣∣ u11 v11 w11 u12 v12 w12 u22 v22 w22 ∣∣∣∣∣∣ = 0. (16) 146 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 Conditions on a Surface F 2 ⊂ En to lie in E4 P r o o f. In the determinant ∆, substitute the expressions for the coefficients of the second quadratic forms (14) to get ∆ = ∣∣∣∣∣∣ ξ3u11 + ξ4v11 + ξ5w11 η3u11 + η4v11 + η5w11 ζ3u11 + ζ4v11 + ζ5w11 ξ3u12 + ξ4v12 + ξ5w12 η3u12 + η4v12 + η5w12 ζ3u12 + ζ4v12 + ζ5w12 ξ3u22 + ξ4v22 + ξ5w22 η3u22 + η4v22 + η5w22 ζ3u22 + ζ4v22 + ζ5w22 ∣∣∣∣∣∣ = 0. This determinant can be written in the form of the product of two determinants ∣∣∣∣∣∣ u11 v11 w11 u12 v12 w12 u22 v22 w22 ∣∣∣∣∣∣ ∣∣∣∣∣∣ ξ3 η3 ζ3 ξ4 η4 ζ4 ξ5 η5 ζ5 ∣∣∣∣∣∣ = 0. The second determinant can not be equal to zero. If the determinant ∣∣∣∣∣∣ ξ3 η3 ζ3 ξ4 η4 ζ4 ξ5 η5 ζ5 ∣∣∣∣∣∣ is equal to zero, then from Eqs. (13) it follows that the normals n1, n2, n3 are linearly dependent. Therefore, Eq. (16) is proved. Equations (15) and (16) allow to construct a number of simple examples of the surfaces in E5, where the plane of the ellipse of normal curvature passes through the corresponding point of the surface. Let the surface be of the form r =   x1 x2 u v w   =   x1 x2 u(x1, x2) v = 1 2(c1x 2 1 + c2x1x2 + c3x 2 2) w = 1 2(d1x 2 1 + d2x1x2 + d3x 2 2)   . We find the derivatives rx1x1 =   0 0 u11 c1 d1   , rx2x2 =   0 0 u22 c3 d3   . Construct two classes of surfaces. For the relations d3 = −d1, c3 = −c1 and u11 + u22=0, we get the Laplace equation rx1x1 + rx2x2 = 0. (17) Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 147 Yu.A. Aminov and Ia. Nasiedkina At d3 = d1, c3 = c1, u11 − u22 = 0, we have the wave equation rx1x1 − rx2x2 = 0. (18) Write the Gauss expansions rx1x1 = Γi 11ri + Lα 11nα, rx2x2 = Γi 22ri + Lα 22nα. From the Gauss equations, the linear independence of the normals nα, and the Laplace equation (17), we get the conditions on the coefficients of the second fundamental forms Lα 11 = −Lα 22, and from the wave equation (18), we obtain the relations Lα 11 = Lα 22, α = 1, 2, 3. From the last two equations and (15) it follows that for both classes of surfaces, the plane of the ellipse of normal curvature passes through a point of the surface. References [1] Yu.A. Aminov, Geometry of Submanifolds. Naukova dumka, Kiev, 2002. [2] R. Curant, Partial Differential Equations. Mir, Moscow, 1964. [3] B.L. Rozhdestvensky and N.N. Yanenko, Systems of the Quasilinear Equations and their Applications to Gas Dynamics. Nauka, Moscow, 1968. [4] E.M. Landis, Some Questions about Qualitative Theory of Elliptic and Parabolic Equations. — Russ. Math. Surv. 14:1 (1959), 21–85. [5] W. Blaschke, Differential Geometry. M–L., ONTI, 1935. [6] A.N. Tikhonov and A.A. Samarsky, Equations of Mathematical Physics. Nauka, Moscow, 1972. [7] S.G. Mikhlin, Course of Mathematical Physics. Nauka, Moscow, 1968. [8] A. Fridman, The Partial Defferential Equations of Parabolic Type. Mir, Moscow, 1968. [9] A. Tychonoff, Theoremes d’unicite pour l’equation de la chaleur. — Mat. Sb. 42 (1935), No. 2, 199–216. [10] I.G. Petrovsky, Lectures on the Theory of Ordinary Differential Equations. Publ. house of the Moscow university, 1984. 148 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 Conditions on a Surface F 2 ⊂ En to lie in E4 [11] I.G. Petrovsky, Lectures on Partial Defferential Equations. Nauka, Moscow, 1961. [12] Yu.V. Egorov and M.A. Shubin, Linear Partial Differential Equations. Foundations of the Classical Theory. — Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. 30 (1988), 5–255. [13] P.K. Rashevsky, Riemannian Geometry and Tensor Analysis. Nauka, Moscow, 1967. Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 149

Conditions on a Surface F² is subset of Eⁿ to lie in E⁴

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