Research Seminar "Differential Geometry"
- The goal of the course is two-fold: ability of practical computations of differential geometry invariants such that Christoffel symbols, curvature tensor etc., as well as deep geometric understanding of underlying concepts and structures
- Ability of manipulating with vector and covector fields in arbitrary local coordinates; computation of commutator of fields and exteriour differential of forms.
- Computation of areas and lengths, finding of principal and Gaussian curvatures of surfaces in the Euclidean three-space
- Ability to compute the curvature of a given metric on the plane.
- Learning methods for finding geodesics and their usage in the local and global study of Rienmann manifolds
- Lerning important examples of Riemann manifolds: sphere, Lobachevski plane in Klein and Poincare models
- Learning different aspects of the notion of connection and interaction between them. Lerning formalism of manipulating with indices and passing from martix to multiindex notations
- Understanding of relationship between the local and global invarians of surfaces
- Vector fields and differential formsManifolds, charts, tangent and cotandent spaces. Vector fields and diffrential forms, coordinate presentation and its transformation under changes of coordinates
- Differential geometry of surfacesThe first and the second quadratic forms, principal curvatures, Gaussian curvature
- Curvature of a plane metricCurvature of a plane metric. Theorema egregium.
- Gauss-Bonnet formula.Connection and curvature forms. Parallel translate of vectors on a surface. Local Gauss-Bonnet formula. Global formula.
- ConnectionConnection as a parralel translate nad connection as covariant derivative. Connection matrix. Guage group transformation. Curvature tensor. Cartan structure equation
- Riemannian manifoldsLevi-Cevita connection. Riemann curvature tensor. Ricci tensor and the scalar curvature. Symmetries of the Riemann tensor
- GeodesicsVariational interpretation. Exponential map. Normal coordinates. Conjugate points. Global geometry of Riemannian manifolds
- Jean-Pierre Demailly. (2007). Complex analytic and differential geometry. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.495EA558
- Differential geometry : connections, curvature, and characteristic classes, Tu, L. W., 2017
- Spivak, M. (1998). Calculus On Manifolds : A Modern Approach To Classical Theorems Of Advanced Calculus. New York: CRC Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=421137