A Phenomenon in Geometric Analysis

It has been established that it is impossible to deduce Euclid V from Euclid I, II, III, and IV. The investigations devoted to the parallel postulate gave rise to a number of equivalent propositions to this problem. Also, while attempting to prove this statement as a special theorem Gauss, Bolyai and Lobachevsky independently found a consistent model of first non-Euclidean geometry namely hyperbolic geometry. Gauss's student Riemann developed another branch of non-Euclidean geometry which is known as Riemannian geometry. The formulae of Lobachevskyian geometry widely used tot sty the properties of atomic objects in quantum physics. Einstein's general theory of relativity is nothing but beautiful application of Riemannian geometry. Einstein derived these field equations by analyzing geometry of space-time. In this study the author re-visited the parallel postulate and by protecting himself under Saccheri's umbrella found a consistent geometric result which challenged the previous contributions in this field.