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Surfaces on time scales and their metric properties
Advances in Difference Equations volume 2013, Article number: 170 (2013)
We present a theoretical framework for surfaces parameterized by the product of two arbitrary time scales. We also study surfaces by delta regular curves lying on them and give their metric tensor known as the first fundamental form with respect to partial delta derivatives.
The theory of time scales, which has recently received a lot of attention, was introduced by Hilger  to unify continuous and discrete analysis. In the study , Bohner and Guseinov introduced the concepts such as curves, delta tangent lines, surfaces and delta tangent planes on time scales. Due to this significant paper, time scale calculus has attracted researchers of differential geometry [3–6]. In the paper , Bohner and Guseinov also gave a brief introduction to surfaces on time scales and discussed integrals on surfaces elaborately.
In this paper, we first study surfaces parameterized by the product of two arbitrary time scales. Since the parametrization lets us obtain a dynamic structure which involves discrete or continuous geometric data, we give the theoretical framework to surfaces in the viewpoint of manifolds. While studying, we only give the results with the delta differential operator, it is straightforward to obtain similar results for a backward differential operator. In , authors briefly introduced that partial delta tangential vectors help us to construct a tangent plane. We also deal with this idea by the surfaces which have less conditions like -complete differentiability. Throughout this paper, we use the chain rule that Bohner and Guseinov presented as follows.
Theorem 1 Let the function f be -completely delta differentiable at the point . If the functions φ and ψ have delta derivatives at , then the composite function
has a delta derivative at the point , which is expressed by the formula
where and .
Theorem 2 Let the function f be -completely delta differentiable at the point . If the functions φ and ψ have delta derivatives at , then the composite function defined by (1) has a delta derivative at the point , which is expressed by the formula
where and .
Detailed proofs of chain rule theorems and significant remarks can be found in . In , vector fields and covariant delta derivatives on time scales are studied for higher dimensional time scales. In Section 3, we present a delta covariant derivative by considering curves lying on the surface. We also study delta integral curves of a vector field which can be derived by a system of dynamic equations on a surface. Finally, in Section 4, we introduce a metric tensor which involves partial delta differentials. By using this tensor, we are able to calculate the length of a delta regular curve lying on the surface with the condition of increasing transformation.
2 Surfaces on time scales
We may consider a surface as a closed subset of , where , , are arbitrary time scales. However, any closed subset of may not be a surface. A theoretical study for any to be a surface is given in this section.
Definition 3 Let be a closed subset of . is a surface if for each point P in , there is a neighborhood A of P and a function , where U is a closed set in and an open set in time scale topology, satisfying the following conditions:
is Δ-differentiable and for all
i.e., φ is Δ-regular.
and is a homeomorphism.
The function is called a surface patch. is called a smooth surface if for all points P in , there exists a surface patch such that .
Speaking about time scale topology, we consider the opens as the sets whose closures are open in the standard real topology. We refer the readers who want to go further into the topic to . Since φ is Δ-regular, one can also conclude that φ belongs to the class .
Proposition 4 Let and let f be a Δ-differentiable function. Then the set
determines a surface.
Proof Assume that the Euclidean coordinate system on a parameter set is . Since coordinate functions t and s are Δ-differentiable and f is also Δ-differentiable, we may conclude that φ is Δ-differentiable. The Jacobian matrix of φ with respect to delta differentiation is
and since for all , , the function φ is Δ-regular. It is also trivial that φ is homeomorphism. □
Theorem 5 Let U and be nonempty closed subsets of and let be a Δ-regular surface patch. If is diffeomorphism, then the function
is a Δ-regular surface patch.
Proof For the function ϕ, let , where and .
First, consider is -completely Δ-differentiable. By the chain rule, we obtain
The constant on the right-hand side of equation (2) is equal to the determinant of the Jacobian matrix
Now, we need to consider the case that the function is -completely Δ-differentiable. If this holds, then we obtain
The constant on the right-hand side of equation (3) is equal to the determinant of the Jacobian matrix .
Example 6 Let and . Since is Δ-regular, the image of defines a surface, and its graph will be as in Figure 1.
Example 7 Let and . It is easy to see is Δ-regular. Therefore the image of defines a surface, and its graph will be as in Figure 2.
2.1 Tangents and Δ-derivatives
The forward tangent line of a Δ-regular curve Γ on time scales is the straight line passing from the point P of the curve through the point , and it has the vector as its direction vector . The same idea can also be extended to the surfaces parameterized by time scales to obtain delta tangent planes .
A natural way to study a surface is via the Δ-regular curves Γ that lie on .
Definition 8 A tangent vector to a surface at a point is the tangent vector at P of a curve in passing through P. The set of all tangent vectors at P is called a tangent space .
Theorem 9 Let be a surface patch of which contains , let be the coordinates of U and let Γ be the Δ-regular curve passing P. Then:
If Γ is -completely Δ-differentiable, then the tangent space at P is spanned by the vectors and .
If Γ is -completely Δ-differentiable, then the tangent space at P is spanned by the vectors and ,
Proof Let Γ be a Δ-regular curve lying on and for .
Suppose that Γ is -completely Δ-differentiable. Then, by the chain rule, we obtain
Therefore, the tangent vector is the linear combination of the vectors and .
Conversely, every vector on can be written in the form of for such constants λ and μ.
Now, let us define
For , i.e., at , the curve Γ is a smooth curve. Therefore, we obtain
This shows that every vector spanned by and is a tangent vector of the curve Γ on at P.
Similarly, assume that Γ is -completely Δ-differentiable. By the chain rule, we get
Therefore, the tangent vector is the linear combination of vectors and .
It is also possible to find such constants and , where every vector on is in the form of .
If we define a smooth curve
then we may obtain
This shows that every vector spanned by and is a tangent vector of the curve Γ on at P.
3 Vector fields and covariant Δ-derivative
Let and denote, respectively, a forward jump operator and a delta operator of for , , where are arbitrary time scales. Let us set
is called an n-dimensional time scale.
A vector field X on is a function that assigns to each point a tangent vector .
Definition 10 Let X be a vector field and . Generally, a vector field is denoted by
where are Euclidean coordinate functions and the set is the natural basis for .
Relationship between a vector field X and its Euclidean coordinate function can be considered as follows: if each of X is -completely Δ-differentiable, then one can say that X is -completely Δ-differentiable.
Suppose that X is a vector field on and . Consider the vector field , where is defined by . It is obvious to see that is a vector field on β. Also, is the closed line parallel to the vector v and .
Definition 11 The vector is called a covariant Δ-derivative of X in the direction of and denoted by .
Now we shall consider Δ-integral curves of vector fields on surfaces. Let be a smooth surface and let X be a vector field on . A curve , where is a nonempty closed set of ℝ, on is called a Δ-integral curve of X if the vector () at each point coincides with the value of X at that point.
It is possible to find equations that should be satisfied by the Δ-integral curve through an arbitrary point . Let be a local coordinate system around P. Then X is locally expressed as
Let be the required Δ-integral curve, and, for the sake of simplicity, choose a parameter such that . If we describe the position of by the local coordinate system as , then we obtain
Thus, the required equation is
The condition of passing through the point P at is expressed as the initial condition .
By the way, (4) is a system of ordinary linear dynamic equations of first order. Theorem 5.8 in  assures us that this system has a unique solution if a coefficient matrix of the system is regressive.
By the existence of the solution, we see that there exists a Δ-integral curve through an arbitrary point P when . We shall consider extending the domain of the integral curve as long as possible to assure the maximality. Since the solution of the system is unique, if two Δ-integral curves pass through the same point at the same time, then they are connected as a single integral curve. So, the way of extending is unique. From this point of view, for each , there exists a Δ-integral curve through the point when , and it cannot be extended any more. This kind of Δ-integral curves are called maximal, and we can conclude that is covered by all the maximal Δ-integral curves which are pairwise disjoint.
4 Metric properties of surfaces on time scales
The metric of a surface is determined by the partial Δ-derivatives of the surface patch φ. Assuming that , the tangent plane to is spanned by the two tangent vectors and . The surface normal vector is orthogonal to both tangent vectors and can be computed as
4.1 First fundamental form
The Jacobian matrix encodes the metric of the surface in a way that it allows measuring transformation of angles, distances and areas by the mapping from the parameter domain to the surface.
Let and be two unit vectors in a parameter space. Cosine of the angle can be computed by the Euclidean dot product . When we consider the dot product of tangent vectors on , we obtain
Therefore, the matrix product induces the metric on a surface on time scales.
Definition 12 Let be a surface on time scales. The first fundamental form of is the assignment to each of the inner product,
i.e., the restriction of the Euclidean dot product to the Δ-tangent space of , and can be shown by I.
Given a surface patch , the matrix for the first fundamental form with respect to the basis is
where , and .
4.2 Length measurement
Since the first fundamental form I defines a metric on a surface, we can measure the length of a Δ-regular curve , defined as the image of a Δ-regular curve in the parameter domain, where for an arbitrary time scale.
Definition 13 Let be a time scale. The function Γ is called a path on if it is increasing. Moreover, the composite function is called a path on the surface, where φ is a Δ-regular surface patch.
Theorem 14 Let be a path on the surface and let be a proper patch in , and suppose that Γ is contained in this patch.
For the coordinate functions and of the path Γ, the length of Γ is
where and is the corresponding delta-differential operator.
Proof Let and , and let their Δ-differential operators be and , respectively. By considering the coordinate changing idea presented in Theorem 5, we can express Γ in terms of coordinates , . Then , hence . By the chain rule, we can obtain
where is the i th component of φ. This shows that and are the components of the Δ-tangential vector with respect to basis .
Now, speed in surface coordinates can be computed as
For length, we have
Hilger S: Analysis on measure chains: a unified approach to continuous and discrete calculus. Results Math. 1990, 18(1-2):18-56. 10.1007/BF03323153
Bohner M, Guseinov GS: Partial differentiation on time scales. Dyn. Syst. Appl. 2004, 13(3-4):351-379.
Guseinov GS, Ozyilmaz E: Tangent lines of generalized regular curves parametrized by time scales. Turk. J. Math. 2001, 25(4):553-562.
Ozyilmaz E: Directional derivative of vector field and regular curves on time scales. Appl. Math. Mech. 2006, 27(10):1349-1360. 10.1007/s10483-006-1007-1
Atmaca SP: Normal and osculating planes of Δ-regular curves. Abstr. Appl. Anal. 2010., 2010: Article ID 923916. doi:10.1155/2010/923916
Kusak H, Caliskan A: The delta nature connection on time scale. J. Math. Anal. Appl. 2011, 375: 323-330. 10.1016/j.jmaa.2010.09.015
Bohner M, Guseinov GS: Surface areas and surface integrals on time scales. Dyn. Syst. Appl. 2010, 19(3-4):435-453.
Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston; 2001.
Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston; 2003.
Atici FM, Guseinov GS: On Green’s functions and positive solutions for boundary value problems on time scales. J. Comput. Appl. Math. 2002, 141(1-2):75-99. 10.1016/S0377-0427(01)00437-X
Bohner M, Guseinov GS: Line integrals and Green’s formula on time scales. J. Math. Anal. Appl. 2007, 326(2):1124-1141. 10.1016/j.jmaa.2006.03.040
Kreyszig E: Differential Geometry. Dover, New York; 1991.
Do Carmo MP: Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs; 1976.
Oberste-Vorth RW: The fell topology for dynamic equations on time scales. Nonlinear Dyn. Syst. Theory 2009, 9(4):407-414.
The authors would like to thank the editor and the referees for their useful comments and remarks.
The authors declare that they have no competing interests.
Authors contributed equally in writing this article. Authors read and approved the final manuscript.
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Atmaca, S.P., Akgüller, Ö. Surfaces on time scales and their metric properties. Adv Differ Equ 2013, 170 (2013). https://doi.org/10.1186/1687-1847-2013-170
- time scales
- delta derivatives
- vector fields
- delta covariant
- first fundamental form