I.Contents：
a) Local differential geometry/tensor calculus：
metric,connection, curvature, geodesics, Jacobi fields, variational formulas, comparision theorems.
b) General theory on vector/fiber bundles:
classification, characteristic classes, homotopy exact sequences, examples.
c) de Rham theory and Hodge theory:
differential forms, de Rham theorem, analytical proof of Hodge theory in Riemannian case.
d) Complex differential geometry:
complex and Kahler manifolds, projective algebraic manifolds, holomorphic curvature, Hodge metrics, line bundles.
*) Selective advanced topics (if the time permits).
II.Course prerequisite：
III.Reference material ( textbook(s) )：
Textbooks:
B.A. Dubrovin, A.T. Fomenko, S.P. Novikov: `Modern Geometry-methods and applications`, vol 1, 2.
Reference books:
B.A. Dubrovin, A.T. Fomenko, S.P. Novikov: `Modern Geometry-methods and applications`, vol 3.
Griffiths/Harris: Principles of algebraic geometry.
IV.Grading scheme：
a) Homework (20-30 %)
b) Final exam (70-80 %)