Differential Geometry (M.Sc.I Sem II Mathematics)

This course deals with the geometry of curves and surfaces.

NEW/REVISED SYLLABUS FOR
M.A. / M. Sc. Mathematics (Part I) (Semester II)
(Introduced from June 2013 onwards)
(i) Paper: MT - 205
(ii) Title of Paper: Differential Geometry
(iii) Specific Objectives: To study the geometry of curves and to develop the calculus
on the surfaces we introduce curves and basic notions of
surfaces.
(iv) A brief note: Theorems and proofs are expected to be prepared from O'Neill, B.:
Elementary Differential geometry, Academic Press,
Revised Edition 2006.
(v) UNIT No. of Lectures
Unit – I Vector space, Euclidean space R3. Tangent vectors and vectors fields, Frame
fields, Natural frame fields, Directional derivative , Curves in R3 and reparametrization
of curves, standard curves, Speed of curve, length of curve. 1- forms, differential forms.
No. of Lectures 15
Unit – II The Frenet Formulae for unit speed curve. Frenet approximation of curves,
Arbitrary speed curves, Frenet formulas for arbitrary speed curve, Covariant Derivative.
Isometries of R3, Orthogonal transformations.
No. of Lectures 15
Unit – III Coordinate patches, surface in R3, simple surface, cylinder surface, surface of
revolution, parametrization of a region, parametrization of cylinder and surface of
revolution, smooth overlapping patches, tangent and normal vector fields on a surface.
No. of Lectures 15
Unit – IV The shape operator of surface M in R3, normal curvature, principal
curvatures, Gaussian and mean curvatures, Umbilic points, fundamental forms of a
surface, computational techniques, special curves on surface, asymptotic and geodesic
curves. No. of Lectures 15
(vi) Recommended Reading : (In MLA/APA Style Sheet Format)
1. Basic Reading: O'Neill, B.: Elementary Differential geometry, Academic Press,
Revised Edition 2006.
References Books:
1. D. Somasundaram: Differential Geometry- First Course, Narosa Publishing
House, New Dehli, 2010.
2. Nirmala Prakash: Differential Geometry, Tata Mcgraw Hill, 1981.
3. K. S. Amur and etl.: Differential Geometry,Narosa Publishing House, 2010.
4. Millman, R. and Parker, G. D. Elements of Differential Geometry, Prentice-Hall
of India Pvt. Ltd. 1977.
5. Hicks, N. : Notes of differential geometry, Princeton University Press (1968)