Surfaces on time scales and their metric properties
© Atmaca and Akgüller; licensee Springer 2013
Received: 15 November 2012
Accepted: 28 May 2013
Published: 17 June 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.
where and .
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.
- i.is Δ-differentiable and for all
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 .
determines a surface.
and since for all , , the function φ is Δ-regular. It is also trivial that φ is homeomorphism. □
is a Δ-regular surface patch.
Proof For the function ϕ, let , where and .
The constant on the right-hand side of equation (3) is equal to the determinant of the Jacobian matrix .
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 .
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 ,
- i.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.
- ii.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 .
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
is called an n-dimensional time scale.
A vector field X on is a function that assigns to each point a tangent vector .
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.
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
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.
Therefore, the matrix product induces the metric on a surface on time scales.
i.e., the restriction of the Euclidean dot product to the Δ-tangent space of , and can be shown by I.
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.
where and is the corresponding delta-differential operator.
where is the i th component of φ. This shows that and are the components of the Δ-tangential vector with respect to basis .
The authors would like to thank the editor and the referees for their useful comments and remarks.
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