Math 765 Riemannian Geometry (Spring 2007)
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Lectures
Lectures are by
Jeff Viaclovsky on Tuesdays and Thursdays
at 9:30-10:45 AM in Van Vleck B211. I will have office hours on Tuesdays
and Thursdays in Van Vleck 803 from 2:15 - 3:15 PM.
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Textbooks
Riemannian Geometry by Peter Petersen.
Differential Geometry Volume I by Spivak.
Riemannian Geometry by Do Carmo.
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Graders
The grader is ?
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Mailing List
The class mailing list is math765-1-s07 "at" wisc.edu.
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Examinations and Homework
There will be some homework assignments, and no exams.
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Brief lecture outline
- Lecture 1: Tuesday, January 23
- Riemannian metrics.
- Length of piecewise smooth curves.
- Distance between points = infimum of lengths of connecting p.s. curves.
- Variation of paths.
- First variation formula of energy functional.
- Lecture 2: Thursday, January 25
- Critical points are smooth, and satisfy geodesic equation.
- Relation between length and energy.
- Exponential map.
- Gauss Lemma.
- Sufficiently short geodesics minimize length.
- Lecture 3: Tuesday, January 30
- Connections in tangent bundle and vector bundles.
- Covariant differentiation along paths.
- Pull-back bundles and induced connection.
- Parallel transport.
- Connection compatible with metric = P.T. is an isometry.
- Lecture 4: Thursday, February 1
- Symmetric connections.
- Fund. Thm. of Riem. Geo.: there exists a unique symmetric
compatible connection.
- Hopf-Rinow and geodesic completeness.
- Lecture 5: Tuesday, February 6
- First variation formula revisited .
- Gauss Lemma revisited.
- Second variation formula.
- Lecture 6: Thursday, February 8
- Curvature tensor.
- Algebraic symmetries of curvature tensor.
- Raising and lower indices (musical isomorphisms).
- Curvature symmetries in coordinates.
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- Lecture 7: Tuesday, February 13
- Curvature tensor in terms of Christoffel symbols.
- Normal coordinates.
- Curvature tensor in normal cooordinates in terms of metric.
- Minimizing geodesic then index Hess E = 0.
- Jacobi operator and Jacobi fields.
- Lecture 8: Thursday, February 15
- Hess E degenerate <-> p and q are conjugate.
- Jacobi fields and variations through geodesics.
- Lecture 9: Tuesday, February 20
- Conjugate points <-> critical points of exp.
- Morse Index Theorem: index counts conjugate points (with multiplicity).
- Lecture 10: Tuesday, February 27
- Fundamental Theorem of Morse Theory.
- Loop space of S^n.
- Non-positive sectional -> no conjugate points.
- Cartan-Hadamard Theorem.
- Topological restrictions on negative curvature.
- Lecture 11: Thursday, March 1
- Sectional curvature determines full curvature tensor.
- Ricci tensor.
- Scalar curvature.
- Myers' Theorem.
- Lecture 10: Tuesday, March 6
- Jacobi fields for constant curvature metrics.
- Expression of constant curvature metric in exponential coordinates.
- Second fundamental form for immersed submanifold.
- Weingarten equation.
- Lecture 12: Thursday, March 8
- Hypersurfaces in Euclidean space.
- Differential of Gauss map = - second fundamental form.
- The Gauss Equations.
- Gauss' Theorema Egregium.
- Totally geodesic submanifolds.
- Riemann's definition of sectional curvature.
- Lecture 13: Tuesday, March 20
- Sturm comparison theorem.
- Jacobi field comparison: K < C, J > J_C.
- Conjugate point comparison, metric comparison.
- Classification of complete, simply connected
spaces of constant curvature.
- Lecture 14: Thursday, March 22
- Group actions, fixed point free, properly discontinuous actions.
- Isometry group of S^n = O(n + 1).
- Spherical space forms, lens spaces, RP^{2n}.
- Isometry group of R^n = E(n).
- Classification of flat surfaces and PSL(2, Z).
- Lecture 15: Tuesday, March 27
- Covariant differentiation in tensor bundle.
- Metric is parallel <-> connection is metric.
- Hessians.
- Differential Bianchi identity.
- Lecture 16: Thursday, March 29
- Algebraic study of curvature tensor.
- Bianchi symmetrization map.
- Ricci contraction.
- Irreducible components of the curvature tensor.
- Lecture 17: Tuesday, April 10
- Curvature under conformal changes of metric.
- Weyl tensor is conformally invariant.
- Hyperbolic space.
- Lecture 18: Thursday, April 12
- Constant curvature metrics conformal to the Euclidean metric.
- Locally conformally flat metrics.
- Thm: n = 3 then lcf iff Cotton tensor vanishes.
- Thm: n > 3 then lcf iff Weyl tensor vanishes.
- Bianchi Identity: divergence (Weyl) = Cotton.
- Lecture 19: Tuesday, April 17
- Lie groups.
- Lie algebras.
- Lie group homomorphism -> Lie algebra homomorphism.
- Lie subalgebras and closed Lie subgroups.
- Lecture 20: Tuesday, April 24
- Lie algebra homomorphism -> local Lie group homomorphism.
- Correspondence between Lie algebras and simply connected Lie groups.
- Left invariant 1-forms.
- Maurer-Cartan equations.
- Existence of maps to G given collections of 1-forms satisfying Maurer-Cartan equations.
- Maps to G differ by left translation <-> same pullback of left invariant 1-forms.
- Lecture 21: Thursday, April 26
- Maurer-Cartan form = canonical Lie algebra valued 1-form.
- Maurer-Cartan equations.
- Restriction of MC form to Lie subgroups.
- MC form for linear groups = g^{-1} dg.
- Homogeneous manifolds G/H and transitive actions of Lie groups.
- Examples of homogeneous manifolds.
- Frame bundle of space forms (S^n, R^n, H^n)
are Lie groups (SO(n+1), ESO(n), O^+(n,1)).
- Lecture 22: Tuesday, May 1
- Orthonormal coframe bundle of a Riemannian manifold.
- Canonical R^n valued 1-form.
- Connection as a Lie algebra-valued 1-form.
- Uniqueness of Riemannian connection.
- Lecture 23: Thursday, May 3
- Existence of Riemannian connection.
- Curvature of a connection.
- Coframe bundle of R^n = EO(n). Maurer-Cartan form = Riemannian connection.
- Flat <-> locally isometric to R^n = Maurer-Cartan equations of EO(n).
- Coframe bundle of S^n = O(n+1). M-C form = Riemannian connection.
- K = 1 <-> locally isometric to S^n = M-C equations of O(n+1).
- K = -1 <-> locally isometric to H^n = M-C equations of O(n,1).
- Cartan's conformal principal frame bundle.
- n =3, Cotton = 0 -> locally conformally flat.
- n > 3, Weyl = 0 -> lcf, proved by Maurer-Cartan equations
of the conformal group O(n+1,1).
- Lecture 24: Tuesday, May 8
- Second fundamental form of distance spheres = Hessian of distance functions.
- Ricatti equation for hessian of distance function.
- SFF for radial metrics and space forms.
- Lecture 23: Thursday, May 10
- Sectional curvature comparison theorem.
- Change of volume element in radial coordinates.
- Bishop's volume comparison theorem.
- Bishop-Gromov volume comparison theorem.
- Laplacian and Hessian comparison theorems.