POINCARE AND NON-EUCLIDEAN GEOMETRY

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1 Bulletin of the Marathwada Mathematical Society Vol. 12, No. 1, June 2011, Pages POINCARE AND NON-EUCLIDEAN GEOMETRY Anant W. Vyawahare, 49, Gajanan Nagar, Wardha Road, NAGPUR , M. S. India. vishwesh.a81@bsnl.in; awv49ngp@yahoo.co.in 1 INTRODUCTION Henry Poincare ( ), a French mathematician and physicist has made major contributions, virtually in all branches of mathematics. He is regarded as a last mathematician to be active over the entire field of mathematics. He originated the study of automorphic functions in complex variables. He was pioneer in topology, astronomy, celestial mechanics, probability theory, physics, and a philosopher too! Poincare was a major prophet of self confident mathematics!! This paper deals with the work of Poincare in non - Euclidean geometry only. Figures are taken from the web sites on Poincare. 2 HISTORY After Gauss ( , German), there was a general thought that there would be no universality in mathematics- like Gauss,- one who was at home in about all branches of mathematics-pure and applied. It was Henry Poincare who proed this view to be wrong. After the demise of Lobachevski ( , Russian), who originated hyperbolic geometry with Bolyai, ( , Hungarian), Poincare pursued the work of these two great mathematicians. At the same time, he was equally influenced by Riemannian geometry. This led him to think of both elliptic and hyperbolic geometry. But he concentrated on hyperbolic geometry, which is richer than elliptic geometry! 3 MOTIVATION In 1879, Poincare made his first important discovery: The inter-relation of non Euclidean geometry with theory of complex functions. Thus, Poincare s most celebrated contribution to mathematics was an automorphic function: A function f in complex variable z is automorphic if it is analytic, except at poles, and is 137

2 138 Anant W. Vyawahare invariant under a linear transformation, defined as : f(z) = (az + b) (cz + d), z = x + iy, where a, b, c, d are constants, real or complex, with ad bc 0. The advantage of automorphic function is that the XY plane can be transformed into the same plane again. That is, an orientation preserving isometry is this function. Thus, a curve C can be mapped into another curve C so that two curves C and C are equivalent. This concept is used in defining the parallels in hyper geometry, and demonstrated in Poincare disc. This type of transformation is called Cremonaian transformation, named after Italian geometer L. Cremona ( ). This isomorphism, in future,played a remarkable role in solutions of differential equations. Poincare showed that just as elliptic and abelian functions are sufficient to integrate differentials, similarly the automorphic functions serve to integrate the linear differential equations with algebraic coefficients. Poincare Disk Under the influence of Lobachevskian geometry, Poincare thought of a simple model of hyperbolic geometry, in which, for every line and a point not on the line, there is more than one parallel line. He devised a model, known as Poincare disk. It consists of all points of interior of a circular disc. The lines of this geometry are arcs inside the circle, with their end points on the circumference of this circle, and perpendicular to it at both ends. The arc endpoints are not the part of lines, but represent the lines at infinity. Now, consider a point P on the disc, and draw the lines (arcs) through P. Many lines can be drawn which are parallel to a fixed line L, and yet none intersects L, so that parallel postulate (fifth) of Euclid is collapsed. Fig. 3.1

Poincare And Non-Euclidean Geometry 139 Poincare Sphere This model is of interest in the view that it shows how a non- Euclidean geometry might arise from the conditions in the physical world.

3 Poincare And Non-Euclidean Geometry 139 Poincare Sphere This model is of interest in the view that it shows how a non- Euclidean geometry might arise from the conditions in the physical world. He suggested another model of Lobachevkian geometry within Euclidean frame work. Suppose that world is bounded by a large sphere of radius R. Consider a point P in the sphere at a distance r from the centre. Then the rays of light (or straight lines ) would not be rectilinear, but would be circles, with radius r, orthogonal to the limiting sphere and would appear to be infinite. Planes would be spheres orthogonal to the limiting sphere, but two such non Euclidean planes would intersect in non Euclidean line. In this case, all axioms of Euclid hold with the exception of the parallel postulate. Fig. 3.2 Poincare makes two assumptions for this sphere: (3.2.1) The temperature at point P is R 2 r 2. (3.2.2) Dimensions of all objects will be in proportion to their temperatures. This universe, which appears bounded to us, will appear unbounded to its inhabitants. The reason is that, as they walk towards the surface of this sphere, their legs- that is their steps - become shorter and shorter so that they will never reach the surface. The shortest distance between two points will be along the arc of a circle joining two points which will be the surface of the sphere at right angles. Half plane A beautiful example of the way the isometries create geometry is a non-euclidean plane of Poincare, known as Half plane. Here it needs to define Isometry. Definition 3.1 A function f on R R is an isometry if d[f(p ), f(q)] = d(p, Q), where d(p, Q), stands for the distance between P and Q. He found a geometry in which there is more than one parallel to a given line through a given point. His plane is the upper half of R R plane, that is, y > 0. This

4 140 Anant W. Vyawahare plane is also called as hyperbolic plane or Poincare upper half plane. Its reflection in complex form is a semicircle above x-axis. The non Euclidean isometries of composites of reflections. Also, Poincare defines the non- Euclidean distances to be equal if there is a non Euclidean isometry carrying one to other. It is important to note that Poincare distinguishes the hyperbolic plane and Euclidean plane in the way the distance is measured. Consider a triangle P QR on hyperbolic plane with their coordinates as P (x, y), Q(x+δx, y), and R(x+δx, y+δy) using infinitesimal calculus. Then, Euclidean length of P R = [δx 2 + δy 2 ], and, Hyperbolic length of P R = [δx 2 + δy 2 ]/y. Obviously, the length of hyperbolic arc is integral of [δx 2 + δy 2 ]/y. within proper limits. Now it follows that in hyperbolic plane, the Euclidean straight line joining two points does not provide the shortest distance between these points. The hyperbolic length is the shortest length on the curve amongst all the curves joining these two points. These shortest distances on hyperbolic pane are known as geodesic segments. These geodesics are either the arcs of Euclidean semicircles that are centered on X axis, or they are the segments of Euclidean straight lines that are perpendicular X axis. Another pleasant property of Poincare half plane is that its angles are ordinary angles; the only difference is that here lines are Euclidean circles,and angles are angles between the circles,( that is, angle between the tangents to these circles). This amounts to saying the perpendicular lines mean the orthogonal circles. Poincare generalizes this half plane model into half space model of non Euclidean space by considering the upper half of three dimensional space,that is, space above XY plane. He defined the isometries of this space to be the products of reflections in spheres with centers on XY plane. The isometries of non Euclidean space can be represented using complex variable z. This because the spheres of reflections are determined by the circles in which they cut XY plane. The isometries determine the half plane (and space also) defined by Poincare. In this way Poincare found that the orient preserving isometries correspond to all the complex automorphic functions f(z), as defined above. Poincare space explains why isometries are linear fractional functions ( as defined above) in all three geometries of the space-euclidean plane,euclidean sphere and non Euclidean space. Poincare has shown that these three geometries occur in non Euclidean space, as, Eucleadian planes are parallel to XY plane, Euclidean spheres as ordinary spheres lying completely above XY plane, and non Euclidean planes as vertical half planes and hemispheres with their centers on XY plane.

5 Poincare And Non-Euclidean Geometry 141 It was Poincare who linked the unification of these geometries. Later an Italian mathematician E. Beltramy ( ) modified the concepts of Poincare by constructing a pseudo-sphere. 4 POINCARE CONJECTURE The Poincare conjecture is a theorem about the characterizations of the three dimensional sphere among three dimensional manifolds. Originally conjectured by Poincare, the claim concerns a space that locally looks like ordinary threedimensional space but is connected, finite in size, and lacks any boundary. Fig. 5.1 The Poincare conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. After nearly a century of efforts by mathematicians, G. Perelman sketched a proof of the conjecture in a series of papers made available in 2002 and ( Perelman was awarded a prestigious Field s Medal, for this work, which he refused to accept). 5 CONCLUSION Henry Poincare was an all-round mathematician, freely touring through all branches of mathematics, with equal genius. His contributions to non-euclidean Geometry is of exceptionally high intellect. In his last letter,poincare wrote: The importance of this subject (hyperbolic geometry) is too great, the geometers who will interest in this subject will be more fortunate than me, they can turn any thing into good.

6 142 Anant W. Vyawahare References [1] Beck, Bleicher and Crowe, Excursions in to Mathematics, University press, Hyderabad, [2] F. Cajori, History of Mathematics, Macmillan and Co., Ltd, USA, [3] John Stillwell, Mathematics and its History, Springer Verlag, NY, USA, [4] Michael Serra, Discovering Geometry, Key Curriculum Press, California, USA, [5] I. Todhunter, Spherical Geometry, Macmillan and Co., Ltd, London, 1932.

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