MA51110: Differential And Riemannian Geometry

MA51110
Course name	Differential And Riemannian Geometry
Offered by	Mathematics
Credits	4
L-T-P	3-1-0
Previous Year Grade Distribution
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Semester	Spring

Syllabus

Syllabus mentioned in ERP

Prerequisite: voidRiemannian space, hypersurfaces, N-ply orthogonal system of hypersurfaces, geodesic, geodesic coordinates, Riemannian coordinates, parallelism of vectors, congruences and orthogonal ennuples, Riccis coefficients of rotation, curvature of a congruence, normal congruences, canonical congruences, Riemann - Christoffel curvature tensor, Schurs theorem, mean curvature, generalized covariant differentiation, Gausss formula, second fundamental form, curvature of a curve in hypersurface, normal curvature, totally geodesic hypersurface, hypersurfaces in Euclidean space and subspaces of a Riemannian space, Dupins theorem, principal normal curvature, lines of curvature, asymptotic directions, conjugate directions (in hypersurface and in subspaces). Parallelism in subspaces, curvature of a curve in a subspace, lines of curvature for a given normal. Manifolds and mappings, differentiable manifolds, submanifolds, tangent space and vector fields, tensors on a vector space, tensor fields, multiplication of tensors, exterior algebra on manifolds, exterior differentiation, integration on manifolds, differentiation on Riemannian manifolds, exterior differential form curvature, principal curvatures, Gaussian and mean curvatures, manifolds of constant curvature.