Geometry of Curves and Surfaces — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

Name. .of the Paper Name of the Corse `: B.Sc. (Honors) Mathematics Notations used are standard. 1. (a) Show that the curve # = a cosh £, y = dr sinh f becomes the hyperbok #2 - 'y2 = 1 and then determine its curvature. (a) Hove that curvature of a curve p(s) is identically 0 if and .only if p(s) (.i) Define the terms .vertex of a plane curve land rotation index of a closed curve. Give an e'xample of a plane curve p(s) with rotation index 0 and a plane curve j5(s) with rotation index`4. Is it posrfule to gradrally deform p(s) to j5(s) ? Justify your answer. (a) Assume that the curvature rof a space ourve p(s) is everywher`e positive. Then .show that the torsion is zero everywhbre if and only if p(s) lies. on a plane. `. (a) Find the curvature and torsion of the hehix ac = a icds, t, y -- a sin t, z = \bt (a > Oh. (c) Define the terin total, curvature of a closed' space curve.. Derive the total curvature.formula for a spaqe curve g(£) (a < £ < a), where f is any parameter (not necessarily the length palam.eter). (a) F6r each of the following surfaces, write the name Of the surface and its I equation in. carte`sian form : Hove that a surface is a plane if its second fundamental form |1 vanishes 4. {a) For .RIght helicoid # = z4 cos zJ, y = zc sin I;, z = flu), determine which part of the sphere is covered by Gauss' spherical map. I (a) Define the exterior, ,differentiation cZ for 0-form,1-form and 2-form. Also, prove that , ddo = 0 when © `is of degree 0. (c) Derive the second structure equation of the surface for Exterior Differential Forms dwa = K01 ^ 02K= 611022-612021 5. (a) Prove that every geodesic of a sphere is a great circle. (b) Determine the connection form wal and the curvature K when a Riemannian metric is given by • dudu + Goudo with respect to the parallel coordinate system. (e) Let C: u = 0 be a geodesic in the construction of the geodesic parallel coordinațe system and g = VG in the Riemannian metric dudu + Gdudu. Prove that gu(0, u) = 0. 6. (a) State Gauss-Bonnet Theorem. Verify it for a sphere with parametric. * = a cos u cos u, y = a cos u sin u, z = a sin u. (b) Prove that : when A is a quadrilateral and l1, l2, lz, l4 are the angles at its vertices. (e) Suppose that A Riemannian metric is defined on a rectangular domain in the (u, v) - plane and its Gaussian curvature is negative or zero. Prove that two geodesics can never intersect at two points such that they together enclose a simply connected domain.