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Introduction
Euclidean Geometry was one of the most important branches of mathematics for the last 2000 years. It was also the main tool used by scientists for the development of astronomy. In 300 B.C., Euclid treated geometry as a deductive system. In 1621, Sir Henry Savile raised some questions concerning what he called “two blemishes” in geometry: the theory of proportion and the theory of parallels. Euclid’s axiom of parallels (postulate V in the first book of Elements) propagates that any two given lines in a plane, when produced indefinitely, will intersect if the sum of two interior angles made by a transversal with these lines is less than two right angles. In 1826, a Russian mathematician, Nicolai Lobachevski, presented a paper to the mathematician faculty of the University of Kazan based on the assumption that it is possible to draw through any point in a plane two lines parallel to a given line. Hungarian mathematician John Bolyai published some results in 1831, which, conceptually, have little difference from those of Lobachevski and perhaps it contains a deeper appreciation of the metric properties of space. However, it was only after Riemann’s profound dissertation on the hypotheses that the underlying foundations of geometry appeared posthumously in 1867, showing the importance of the metric concepts in geometry.
Riemann appeared to have been unaware of the work of Lobachevski and Bolyai, although it was well known to Gauss. Later, Beltrami published his classical paper on the interpretation of non-Euclidean geometries (1868) in which he analyzed the work of Lobachevski, Bolyai, and Riemann and stressed the fact that the metric properties of space are mere definitions. It appears that three consistent geometries are possible on surfaces of constant curvatures: Lobachevskian on a surface of constant negative curvature, Riemannian on a surface of constant positive curvature, and Euclidean on a surface of zero curvature. These geometries are called hyperbolic, elliptic, and parabolic, respectively.
Tensor calculus is concerned with the study of abstract objects, called tensors, which are independent of frames of reference used to describe them. In the early part of the 17th century, geometry was developed by French mathematician Descartes. The great physicist W. Voigi discovered tensors and gave them this name in his remarkable book A Textbook of Crystal Physics published in 1910. Ricci and his student, Levi-cita, (1901) have been developing the subject of tensors, but it is well known that Einstein’s use of tensors as a tool in his general theory of relativity (1914) was mainly responsible for the sudden emergence of tensor calculus as a popular field of mathematical activities. The application of tensors in the field of mathematics and physics was mainly accelerated after the publication of Einstein’s famous paper The General Theory of Relativity.
Tensor is a generalization of the term vector and Tensor Calculus is a generalization of vector analysis. The concept of invariance of mathematical objects, under coordinate transformations, permeates the structure of tensor analysis to such an extent that it is important to get at the outset a clear notion of the particular brand of invariance we have in mind. In the given reference frame, a point P is determined by a set of coordinates, xi. If the coordinate system is changed, point P is described by a new set of coordinates, but the transformation of the coordinates does nothing to the point itself.
Here, we discuss how the term tensor may be considered as a generalization of the term vector.
We start with a two-dimensional Euclidean space, E2, provided with a system of rectangular Cartesian coordinates. Let P and Q be two points of E2, with (x1, x2) and (y1, y2) as their respective coordinates. Then, the coordinates of the directed line segment are (x1 − y1, x2 – y2).
Denoting x1 − y1 and x2 – y2 by z1 and z2 respectively, the coordinates of can be expressed as
(0.1)
Next, we consider an orthogonal transformation of coordinate axes given by
where is orthogonal, with its determinant equal to 1.
We can express (0.2) as follows:
denoted by (x1, x2) and (y1, y2), the coordinates of Q and P in the new coordinates system. Then, the coordinates of vector in the new coordinate system are and are denoted by Then can be expressed as
Hence, Equation (0.4) can be written as
(0.5)
[from Equation (0.3) we get
The above equation shows that the coordinates of a vector of E2 transform according to a certain law in Equation (0.6) when referring to a new coordinate system.
This was first pointed out by Felix Klein in 1872.
Similarly, if we consider n vectors, it can be shown that there exists an object with components in a coordinate system, according to
(0.7)
Hence, we conclude that a tensor of E2 may be regarded as generalization of a vector of E2 defined from the transformation law. Similarly, a tensor of En can be obtained as a generalization of a vector of En. A tensor obtained from the orthogonal transformation of a rectangular Cartesian coordinate system is called a Cartesian Tensor and a tensor from a general transformation coordinate system is simply called a tensor.
The flourishing of the subjects of tensors and Differential Geometry and Mechanics is due to Einstein and Grassman. Then, many mathematicians and researchers developed Differential Geometry with tensor applications.
Sasaki and Hsu defined and studied almost all contact structures and their integrability conditions. In 1970, Yano and Okumura studied structure manifold and Walker, A.G. (1955) studied the properties of the manifold (λu, v,) with an almost product structure in which there exists a (1,1) tensor field, f, whose square is unity. K. Yano (1963) generalized the concept of an almost complex structure and defined f-structures as a (1,1) tensor field f (satisfying f3 + f = 0). In 1972, K. Kenmotsu studied a certain class of an almost contact manifold. Janssen and Vanhecke (1981) named this structure a Kenmotsu structure and the differentiable manifold equipped with this structure is called a Kenmotsu manifold. Many authors have studied slant immersions in almost Hermitian manifolds. The study of Differential Geometry of tangent and cotangent bundles was started by Sasaki (1958) and then Yano and Davies and Ledger. The theory of submanifolds as a field of Differential Geometry is as old as Differential Geometry itself. A study of the submanifolds of a manifold is a very interesting field of Differential Geometry. In 1981, B.Y. Chen, D.E. Blair, A. Bejancu, M.H. Sahid (1994-95), and some others studied different properties of submanifolds. Sasaki (1960) and others studied differentiable manifolds in detail.
Differential Geometry is the study of geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of Tensor Calculus. For the study of curve by this method of calculus, its parametric representation is a covariant and discuss tangent and normal and binormal, which is of fundamental importance to the theory of the curve. We will study the geometric properties of surface imbedded in the three-dimensional Euclidean space by means of Differential Geometry, termed as intrinsic properties and intrinsic geometry of surface. The study of the geometry of surfaces was carried out from the point of view of a two-dimensional being whose universe is determined by the surface parameters u1 and u2 and it was based entirely on the study of the first quadratic differential form.
Differential Geometry has a long and rich history and, in addition to its intrinsic mathematical value and important connections with various other branches of mathematics, it has many applications in various physical sciences, e.g., solid mechanics, computer tomography, and general relativity. Differential Geometry is a building block in Physics and Classical Mechanics, which was developed extensively by Newton. It deals with the motion of particles in a fixed frame of reference. Within those frames, other coordinate systems may be used so long as the metric remains Euclidean. The reference system generally used in astronomy is determined by “fixed stars”. It is termed as the primary inertial system. The motion of the earth relative to its primary inertial system is so negligible that Newtonian laws which can be applied without modification to the study of motion of particles is referred to as a system of axes fixed in the earth.
