Introduction to Differential Geometry

computer science
mathematics
physics

elective courses

Linear Algebra, Multvvariable Calculus, basic Topology

Classroom

Basic differential geometry: submanifolds of euclidean spaces and tangent vectors, curves and moving frames. Frenet-Serret theorem, cuvature and torsion of curves in 3-dimensional space. Surfaces in 3-dimensional space., I and II fundamental forms, principal curvatures, Gauss curvature. Theorema egregium and intrinsic geometry of surfaces. Geodesic curves on surfaces. Covariant derivative of vector fields and parallel transport. The Gauss-Bonnet-Theorem. Abstract Riemannian manifolds. Models of the hyperbolic plane.

1. Submanifolds of euclidean spaces, parametrizations and charts, constant rank theorem. Tangent vectors and tangnet spaces. Smooth maps and their differentials. Basis of the tangent spaces determined by parametrizations . Linear groups as manifolds. External and intinsic isometries of submanifolds. Isometires of eucildean spaces.

2. Moving frame method. Curves in euclidean spaces; in particular 2 and 3-dimensional.

3. The Frenet-Serret equations, as applictaion of the moving frame theorem. Existence and uniqueness up to external isometry of a ciurve with prescibed cirvatures. Umlaufsatz (info).

4. Oriented surfaces in 3-dimensional euclidean space. The Darboux frame of a curve on the surface - normal and geodesci curvature, geodesic torsion. Geodesic curves. Geometric interpretation of the normal curvature as cuvature of a plane curve. The Weingarten map, proncipal curvature, the Gauss curvature, mean curvature. I and II fundamental forms and their coefficients with repsect to parametrizations.

5. Vector fields along curves on surfaces and their covariant derivatives. Parallel vector firlds and parallel transport of tangent vectors. Local Gauss-Bonnet theorem.

6. Riemannian metrics on open subsets of affine spaces. Lenght of curves, measure determined by a metric. Geodesics. Models of the hyperbolic plane.

1. C. Bowszyc, J. Konarski, Wstęp do geometrii różniczkowej, Wydawnictwa Uniwersytetu Warszawskiego, Warszawa 2016.

2. M. Do Carmo, Differential geometry of curves and surfaces. Revised & updated second edition, Dover Publications, Inc., Mineola 2016.

3. A. Gray, E. Abbena, S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica. Third Edition, Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton 2006.

4. S. Jackowski, Geometria różniczkowa. Pomocnik studenta, Skrypt MIM UW, Warszawa 2018. – dostęp ze strony www autora.

5. W. Klingenberg, A course in differential geometry, Springer-Verlag, New York-Heidelberg 1978.

6. S. Montiel, A. Ros, Curves and surfaces. Second edition, Graduate Studies in Mathematics 69, American Mathematical Society, Providence; Real Sociedad Matemática Espanola, Madrid 2009.

7. J. Oprea, Geometria różniczkowa i jej zastosowania, PWN, Warszawa 2002

A student:

1. Understands notions of a submanifold, tangent vectors and differential of a smooth map.

2. Understands geometric sense of the normed parametrization of a curve, and of its curvature and torsion.

3. Understands difference between external and intrinsic properties of submanifolds.

4. Is able to recognize points of negaitive, ositive and zero Gauss curvature on a surface and knows that the Gauss curvature is an intrinsic invariant.

5. Is able to give examples of geodesics and knows that they are invariant under isometries.

6. Is able to describe examples of the parallel transport.

7. Knows examples of the constant curvature surfaces and properties of geodesic triangles.

8. Understands gemeotric sense of the local Gauss-Bonnte theorem and topological sense of the global version.

9. Knows examples of the local abstract Riemannian manifolds in paricular hyperbolic plane.

Final grade based on an essay and written exam consisting of quiz and problems.