微分幾何是高維度空間的研究中最重要的理論，也是廣義相對論以及部分理論物理學的數學基礎。本課程旨在提供足夠的理論，使同學修完一年的課程之後，有能力學習更專業的幾何研究課程如辛幾何，複幾何，低維度微分拓樸或將微分幾何應用在其他學科上。基本課程包含

1. Differentiable Manifolds: Partition of unity, tangent spaces, vector fields and flows, Lie derivatives, Frobenius integrability theorem, Sard’s theorem, Whitney imbedding theorem.
2. Tensor Calculus: Lie derivatives of tensors, Cartan’s theory on differential forms, Stokes theorem, de Rham cohomology, de Rham theorem. Introduction to the Hodge theory of harmonic forms.
3. Riemannian Geometry: Levi-Civita connection, geodesics, exponential map, Riemann curvature tensor, 1st and 2nd variations of arc length, Hopf-Rinow theorem, Bonnet-Synge theorem, Jacobi fields, Cartan-Hardamard theorem, Cantan-Ambrose-Hicks theorem.

下學期若尚有剩餘的時間，則可能著重於某些專題，如以下之一：

1. Vector Bundle Theory: Connections and curvature, Chern-Weil theory on Chern classes, Spinors and Dirac operators, the Atiyah-Singer index theorem.
2. Geometric Analysis: Comparison geometry, Bochner principle, eigenvalue estimates, analysis on complete manifolds, introduction to Ricci flows.
3. Minimal Submanifolds: 1st and 2nd variations, Weierstrass representations, Platean problem, introduction to calibrated and Kaehler geometry.

以下為常用之參考文獻：(前三本為常用教科書)

• Warner: Foundations of Differential Manifolds and Lie Groups.
• Do Carmo: Riemannian Geometry.
• Jost: Riemannian Geometry and Geometric Analysis.
• Berline et. al.: Heat Kernels and Dirac operators.
• Li: Lecture Notes on Geometric Analysis.
• Lawson: Lectures on Minimal Submanifolds.
• Schoen and Yau: Lectures on Differential Geometry.