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\title{Math647: Differential Geometry \\
Fall 2002 \\
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\noindent
{\bf TIME:} \hspace{5pt}
MWF: 3:00 - 3:50.
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\noindent
{\bf INSTRUCTOR:} \hspace{5pt}
Xiaobo Liu.
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\noindent
{\bf COURSE DESCRIPTION:}
The first part of this course will be an introduction to Riemannian
geometry, which studies geometric properties of differentiable
manifolds equipped with positive definite inner products on
tangent spaces. The distance function, covariant derivative, and
curvature tensor associated to a Riemannian metric will be studied.
The second part of this course studies complex manifolds, i.e. those
differentiable manifolds locally look like a domain in $C^{n}$.
Topics include Hermitian metrics, Chern classes, and other structures
of complex manifolds.
The intersection of Riemannian geometry and complex geometry is the
K\"{a}hler geometry where the complex structure is compatible with the
Riemannian metric. If time permits, we will also cover basics of
K\"{a}hler geometry.
Although basic concepts of differentiable
manifolds, differential forms, tensors, vector bundles will be briefly
reviewed. Prior knowledge on these concepts will surely be very helpful.
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{\bf REFERENCE:}
Fangyang Zheng
{\it Complex Differential Geometry}, International Press, 2000.
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