GEOMETRY AND TOPOLOGY
AIM OF THE CLASS
This class is a regular introductory class to topology and differential geometry, as well as preparation for the
preliminary examination Geometry and Topology which is a part of the doctoral study of mathematics.
Upon successful completion of this class, students will be able to:

show knowledge and understanding of concepts and results of general and algebraic topology, as well as
differential and Riemannian geometry.
CLASS CONTENT
General topology
Metric, normal, compact and paracompact spaces. Urysohn's lemma. Tietze's extension theorem. partition of
unity. Tychonoff's theorem. Baire's category theorem.
Fundamental group and coverings
Homotopy and homotopy type. Fundamental group. Van Kampen's theorem. Covering spaces.
Homology
Simplicial and singular homology. Homotopy invariance. Exact homology sequence and excision. Axioms.
Some applications. Cohomology.
Diferential manifolds
Definition and examples. Tangent and cotangent bundles. Vector fields and differential forms. Algebra of
differential forms. Submanifolds. Orientation. Riemannian manifolds. Isometries.
Riemannian geometry of 2-manifolds
Parallel transport. Connections. Cartan's structural equations. Curvature. Riemannian connections. Geodesic
curves. Spaces of constant curvature.
Surfaces in R3
Gauss map. Weingarten map. Interpretation of the parallel transport, Riemannian connections and curvature for
2-manifolds embedded in R3. Second fundamental form.
REQUIRED LITERATURE
G. E. Bredon. Topology and Geometry, Springer, 1993.
V. Guillemin, A. Pollack. Differential Topology, Prentice-Hall, 1974.
A. Hatcher. Algebraic Topology, Cambridge University Press, 2002.
http://www.math.cornell.edu/~hatcher/AT/ATpage.htm
J. M. Lee. Introduction to Smooth Manifolds, Springer, 2000.
J. R. Munkres. Topology. Second Edition, Prentice Hall Inc., 2000.
I. M. Singer, J. A. Thorpe. Lecture Notes on Elementary Topology and Geometry, Springer, 1967.
ADDITIONAL LITERATURE
J. M. Lee. Riemannian Manifolds, An Introduction to Curvature, Springer, 1997.
W. S. Massey. A Basic Course in Algebraic Topology, Springer, 1991.
F. W. Warner. Foundations of Differential Manifolds and Lie Groups, Springer, 1983.