瞭解Riemannian manifolds, Riemannian Metrics, Riemannian Connections, Geodesics 及 Curvature 等基本概念，並探討這些概念在矩陣理論（特別是正定矩陣空間space of positive definite matrices）以及Lie groups上的應用。

Understand the basic concepts such as Riemannian manifolds, Riemannian Metrics, Riemannian Connections, Geodesics and Curvature, and explore the application of these concepts in matrix architecture (especially positive definite matrices space of positive definite matrices) and Lie groups.

In this course we will introduce some important notions in metric geometry and Riemannian geometry such as Length Spaces, Hyperbolic Space, Spaces of Bounded Curvature, Alexandrov Spaces, Curvature of Riemannian Metrics, Space of Metric Spaces, Gromov-Hausdorff distance etc.

In this course we will introduce some important notions in metric geometry and Riemannian geometry such as Length Spaces, Hyperbolic Space, Spaces of Bounded Curvature, Alexandrov Spaces, Curvature of Riemannian Metrics, Space of Metric Spaces, Gromov-Hausdorff distance etc.

1. M. P. Do Carmo, Riemannian geometry, Birkhauser Boston, 1992.
2. R. Bhatia, Positive denite matrices, Princeton University Press, 2009.
3. D. Petz, Matrix Analysiis with some Applications, http://bolyai.cs.elte.hu/~petz/matrixbme.pdf.
4. F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups (Graduate Texts in Mathematics) (v. 94), Springer

1. M. P. Do Carmo, Riemannian geometry, Birkh auser Boston, 1992.
2. R. Bhatia, Positive de nite matrices, Princeton University Press, 2009.
3. D. Petz, Matrix Analysiis with some Applications, http://bolyai.cs.elte.hu/~petz/matrixbme.pdf.
4. F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups (Graduate Texts in Mathematics) (v. 94), Springer