Course Description: Riemannian geometry is the study of smooth manifolds equipped with a Riemannian metric. With this additional structure one is able to define geometric concepts such as geodesics, connections and curvature.  The Riemannian geometer frequently explores the relationship between these concepts and the topological and differentiable structures of the underlying manifold. For example, the celebrated sphere theorem tells us that if M is a  compact, simply-connected, smooth manifold of dimension n that admits a Riemannian metric g with sectional curvature satisfying 1/4 < Sec_{(M, g)} <= 1, then M is homeomorphic to Sⁿ. As a corollary we can conclude that the universal cover of any compact manifold which admits a Riemannian metric g with 1/4 < Sec_{(M, g)} <= 1 must be a sphere.

This course will serve as an introduction to this classical and vibrant area of research with an aim towards developing both theoretical and computational proficiency. It should be of relevance to students with interests in geometry, topology and (mathematical) physics. The topics covered will include some of the following.

• Riemannian Basics: Riemannian metrics; Affine & Levi-Civita Connections (Christoffel Symbols); geodesics, the exponential map, and the geodesic flow; curvature tensor, sectional curvature, Ricci & Scalar curvature; Curvature on compact Lie groups; Killing Fields.
  
• Computations: Computation of Riemannian basics on surfaces of revolution, Sⁿ, Lie groups, RPⁿ, CPⁿ and other homogeneous spaces, and warped products.
• Geometry of Isometric Immersions: Immersions; The second fundamental form; Principal directions & curvatures; totally geodesic submanifolds; Sectional Curvature & the Gaussian curvature of surfaces; Minimal surfaces; The equations of Gauss, Ricci and Codazzi (i.e., the fundamental eqs.).
• Jacobi Fields \& Interpretations of Curvature: Jacobi equation and Jacobi Fields; Curvature and the spreading of geodesics; Conjugate points and singularities of the exponential map.
• Variations of Energy & Geodesics: 1^st and 2^nd Variation Formulas; Geodesics are locally minimizing; Theorems of Bonnet-Meyers and Synge.
• Complete Manifolds & the theorems of Hadamard and Hopf-Rinow
• Spaces of Constant Curvature: Theorem of Cartan on Curvature and Metrics; Space forms \& totally discontinuous group actions.
• Manifolds of Negative Curvature: Existence of closed geodesics; Preissman's theorem; Non-existence of Killing fields; Ergodicity of the geodesic flow (optional).
• Sphere Theorem
• Isometry Groups & Isometric Actions: Isometry groups as Lie groups; Killing fields; group actions; action/fundamental vector fields.
• The Geometry of Riemannian Submersions: submersions; the A- and T-tensors; O'Neill's formula; Principal Bundles \& Connections.
• The Geometry of Lie Groups and other Homogeneous Spaces
  

I will announce the definitive grading guide in the coming weeks.

• Topics: Riemannian Metrics, Isometries , Lie groups & homogeneous spaces (definitions)
  
• Reading (Due Next Time):  Chp. 1 & 2

• Topics: Review of Smooth manifolds. We'll use this hour to help people new to manifolds catch up.
• HW 1 Due Today
  

• Topics: Flat tori; Lie Groups, Lie Algebras & One-parameter subgroups
  
• Reading (Due Next Time): Chp. 2
  

• Topics: Lie Algebras & One-parameter subgroups; Killing Fields;  Connections
  
• Reading (Due Next Time): Chp. 2 & 3
  
• HW2 (Due April 20)
  

• Topics: Connections & Parallel Transport
  
• Reading (Due next Time): Chp. 2 & 3

• Topics: Parallelism; Torsion and Curvature Tensor of a Connection, Flat Connections; Levi-Civita Connection
  
• Reading (Due Next Time): Chp. 2 & 3

• Topic: Levi-Civita Connection & Geodesics
  
• Reading (Due Next Time): Chp. 3 & 4
  
• HW3 (Due April 27)

• Topic: More on Geodesics: normal neighborhoods & minimizing properties
  
• Reading (Due Next Time): Chp. 4

• Topic: Minimizing properties of geodesics; Riemannian submersions & group actions
  
• Reading (Due Next Time): Chp. 4
• HW4 (Due May 9)
  

• Topic: Riemannian submersions, horizontal lifts and geodesics; Curvature: covariant derivative of tensors, curvature tensor, sectional curvature
• Reading (Due Next Time):

• Topic: Riemannian Submersions & Sectional Curvature: the A-tensor and O'Neills' formula
  
• Reading (Due Next Time): Chp. 5
  

• Topic: Sectional Curvature of CP^n; Ricci and Scalar Curvature, the curvature operator; Jacobi Fields
  
• Reading (Due Next Time): Chp. 5 & 9 (Note: You should start reading Chps. 7 & 8. We will most likely not cover them in class, but you should be able to understand them on your own.)

• Topic: Jacobi Fields
• Reading (Due Next Time): Chp. 9 (Note: You should start reading Chps. 7 & 8. We will most likely not cover them in class, but you should be able to understand them on your own.)

• Topic: Jacobi Fields and Conjugate Points,  Geodesics w/o conjugate points
  
• Reading (Due Next Time): Chp. 9 (Note: Read Chps. 7 & 8. We will most likely not cover them in class, but you should be able to understand them on your own.)
  

• Class: Variations of Energy & the Index Form
• Reading (Due Next Time): Chp. 6 (Note: Read Chps. 7 & 8. We will most likely not cover them in class, but you should be able to understand them on your own.)
• HW5 (due May 30)
  

• Topic: The Theorems of Bonnet-Meyers, Weinstein and Synge
• (Due Next Time): Chp. 6 (Note: Read Chps. 7 & 8. We will most likely not cover them in class, but you should be able to understand them on your own.)

• Topic: Isometric Immersions and the Second Fundamental Form
  
• Reading (Due Next Time): Chp. 6 (Note: Read Chps. 7 & 8. We will most likely not cover them in class, but you should be able to understand them on your own.)

• Topic: No Class
• Reading (Due Next Time):

• Topic: Isometric Immersions and the Second Fundamental Form

• Topic: Geometry of Lie Groups 
  
• Reading (Due Next Time):