Some contributions to the geometry of riemannian and lorentzian manifolds

Abstract

Geometry is the very essential category of mathematics related to the properties, newlinerelationships and measurement of points, lines, curves and surfaces. The determination newlineof calculus in the 17th century opened up the study of more complexed plane curves newlinesuch as those produced by the French mathematician Ren Descartes (1596-1650) with newlinehis compass . In particular, integral calculus provides general solutions of the ancient newlineproblems of finding the arc length of plane curves and the area of plane pictures. This newlineassist to the examination of curves and surfaces in space. Originally, a body of practical newlineknowledge regarding lengths, areas and volumes in the third century B.C., geometry was put into an accepted form by Euclid, whose exploration Euclidean geometry set a standard for many centuries to follow. newlineDifferential geometry is most likely as old as any mathematical discipline is concerned newlineand definitely was well projected after Newton and Leibnitz have set down the establishment newlineof calculus that survey the geometry of curves, surfaces and manifolds. In the newlinebeginning of 19th century, Gauss laid down the foundation of differential geometry of surfaces newlinein three dimensional Euclidean space. The notion of differential geometry of more newlinethan three dimensions was introduced by Riemann (1854). Differential Geometry is very newlinemuch what the name implies: geometry done using differential calculus. newline newline