Differential Geometry I 1000-134GR1
The program of the lecture.
1. Differentiable manifolds: maps, atlases, paracompactness, the partition of unity. Differentiable maps, submanifolds, Examples: surfaces in R^3, quotients (tori, real projective space), complex projective spaces, Lie groups.
2. Tangent space: tangent vectors as directions of curves and as derivations. Tangent maps of differentiable maps of manifolds. Tangent bundle, cotangent bundle, vector bundles. Sections of bundles, vector fields, Lie bracket. Differential forms, algebra of differential forms, exterior derivative.
3 a. Oriented manifolds, manifolds with boundary, Stokes theorem.
b. One-parameter groups of diffeomorphisms, integration of vector fields, Lie derivative of vector fields, the connection with Lie bracket. Frobenius theorem.
4. Riemann manifolds, curves on them (length, angles), parallel transport, geodesics. Examples: surfaces in R^n, Lobachevski plane.
5. Surfaces in R^3: Gauss map, Weingarten map, II form, main curvatures, Gauss curvature, Christoffel coefficients, theorema egregium. Curves on surfaces, geodesic curvature, parallel transport, geodesic curves, defect of triangle. Gauss-Bonnet theorem.
Type of course
Learning outcomes
The student
1. knows a definition of a smooth manifold and a submanifold, coordinate chart and atlas. Can analytically describe the construction of the atlas for a sphere, torus and manifolds related with problems of linear algebra (projective spaces, Grassmannians). Can define projective spaces, a torus and Klein's bottle with the help of product of spaces.
2. knows definitions of smooth mapping, immersion, submersion, diffeomorphism and submanifold. Can find a set of singular points and values of mappings.
3. Knows the definition of a tangent vector and a differential form at a point, and of the vector field and the differnetial form as a section of the tangent and cotangent bundles respectively. Understands the equivalence of definitions of a tangent vector as an equivalence class of parametrised curves at a point and as differentiation of the algebra of smooth functions at a point. Can find an affine tangent space to a submanifold of the affine space. Understands the connection between the Lie bracket of vector fields and the structure of the Lie algebra of the classical Lie groups. Knows the Frobenius theorem.
4. Can find the normal vector field of an oriented hypersurface parametrised and defined by equations, and apply the Stokes theorem.
5. Knows the definition of a smooth partition of unity, construction of Riemannian metrics on paracompact manifolds, and of the first fundamental form on submanifolds of the Euclidean space. Understands the connetion between the inner and outer geometry of the submanifolds in the Euclidean space and the Egregium theorem.
6. Can compute the angle between curves and the length of the curve on a Riemannian manifold, find geodesics and parallel transport of a vector along a curve. Knows the Poincare and Klein models of the non-Euclidean geometries.
7. Knows the definition of the Euler characteristics of a surface and
can compute it by using a triangulation. Can apply the Gauss-Bonnet theorem and compute the areas of geodesic polygons on the surfaces of constant curvature.
8. Understands the connection of local indices of vector fields and Euler characteristics and can apply the Poincaré-Hopf theorem.
Bibliography
1. T.Aubin, A course in Differential Geometry, AMS 2001
2. J.Oprea, Differential Geometry and its Applications, Prentice Hall, 1997,
Additional information
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