Topics in Differential Geometry

We consider a Lorentz transformation in which the moving frame moves with uniform velocity in an arbitrary direction. Transformation matrix for such a transformation is found in the book of Misner et al and also that of Weinberg. We use this matrix to show that the volume element and a differential operator are invariant. We use these in turn to define other differential operators in an invariant manner. These operators are used in the study of electromagnetism. Another topic that may find application to topics in physical sciences is ‘Irrotational vectorfields on a Surface in Euclidean space’. It is shown that parallel surfaces preserve such a vectorfield and also that every real valued differentiable function on a sphere generates an irrotational vectorfield. A change of metric of a 2-dimensional Riemannian manifold leads to an invariant tensorfield, the vanishing of which leads to the conclusion that the geometrical object represented by the manifold is a torus. A parametric hyper-helicoid with Weierstrass type representation is another significant topic discussed in some detail.