Riemannian Geometry
A.Y. 2026/2027
Learning objectives
The course aims to introduce students to advanced topics in Riemannian geometry that are fundamental for current research in the field. In particular, various techniques will be developed to establish connections between the curvature and the topology of a manifold and its submanifolds.
Expected learning outcomes
A solid foundation in Riemannian geometry; familiarity with tools from Geometric Analysis (including distance function analysis and comparison theorems, function theory, differential forms, and variational methods); and the ability to use these tools to obtain topological information about a Riemannian manifold and its submanifolds.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Lectures are in person. In case of emergency, lectures will be given in online, syncronous mode.
Course syllabus
QUICK REVIEW OF RIEMANNIAN GEOMETRY
- Metric, connection, curvature.
- First variation of energy: geodesics, exponential map, normal chart.
- Hopf-Rinow's theorem. Completeness and coverings.
- Second variation of energy and Jacobi fields. Conjugate points and their properties.
- Maximal domain of a normal chart, cut-locus. Regularity of the distance function.
SUBMANIFOLD THEORY
- Isometric immersions. Induced connection, second fundamental form and mean curvature. Fundamental equations. Gauss map. Hadamard's theorem for hypersurfaces.
- First variation of the area and minimal submanifolds (hints, time permitting)
This first part will be, at least partly, developed in an interactive, problem-solving mode, to familiarize with the basic tools of the discipline.
COMPARISON THEOREMS AND APPLICATIONS
- Hessian comparison theorem for the distance function
- Applications: theorems by Cartan, Tompking (and Preissman, time permitting). Length comparison.
- Laplacian comparison theorem
- Applications: Bonnet-Myers and Cheng's diameter rigidity theorem.
- Bishop-Gromov's volume comparison.
SPLITTING THEOREM:
- Rays and lines. Busemann function and Cheeger-Gromoll's splitting theorem.
- Applications: structure of the universal covering of a compact manifold with non-negative Ricci curvature.
HODGE THEORY
- Hodge star-operator, Harmonic forms
- Hodge theorem (without proof)
- Application: Betti number estimates for manifolds with non-negative Ricci curvature (Bochner' technique)
NONSMOOTH CRITICAL POINT THEORY
- Regular and critical points of the distance function, c-pseudogradienti
- Nonsmooth deformation lemma.
- Application: disc theorem.
- Toponogov's theorem
- Application: Grove-Shiohama's theorem (diameter sphere theorem)
- Soul Theorem
- Metric, connection, curvature.
- First variation of energy: geodesics, exponential map, normal chart.
- Hopf-Rinow's theorem. Completeness and coverings.
- Second variation of energy and Jacobi fields. Conjugate points and their properties.
- Maximal domain of a normal chart, cut-locus. Regularity of the distance function.
SUBMANIFOLD THEORY
- Isometric immersions. Induced connection, second fundamental form and mean curvature. Fundamental equations. Gauss map. Hadamard's theorem for hypersurfaces.
- First variation of the area and minimal submanifolds (hints, time permitting)
This first part will be, at least partly, developed in an interactive, problem-solving mode, to familiarize with the basic tools of the discipline.
COMPARISON THEOREMS AND APPLICATIONS
- Hessian comparison theorem for the distance function
- Applications: theorems by Cartan, Tompking (and Preissman, time permitting). Length comparison.
- Laplacian comparison theorem
- Applications: Bonnet-Myers and Cheng's diameter rigidity theorem.
- Bishop-Gromov's volume comparison.
SPLITTING THEOREM:
- Rays and lines. Busemann function and Cheeger-Gromoll's splitting theorem.
- Applications: structure of the universal covering of a compact manifold with non-negative Ricci curvature.
HODGE THEORY
- Hodge star-operator, Harmonic forms
- Hodge theorem (without proof)
- Application: Betti number estimates for manifolds with non-negative Ricci curvature (Bochner' technique)
NONSMOOTH CRITICAL POINT THEORY
- Regular and critical points of the distance function, c-pseudogradienti
- Nonsmooth deformation lemma.
- Application: disc theorem.
- Toponogov's theorem
- Application: Grove-Shiohama's theorem (diameter sphere theorem)
- Soul Theorem
Prerequisites for admission
A basic course in Differential Geometry (roughly corresponding to the first 5 chapters of Do Carmo's book in the bibliography).
A basic knowledge of the following topics is useful but not strictly necessary.
- PDE (just comparison and maximum principles for harmonic functions, notion of weak solutions of an elliptic PDE);
- definition of deRham's cohomology and deRham's Theorem;
- covering spaces.
A basic knowledge of the following topics is useful but not strictly necessary.
- PDE (just comparison and maximum principles for harmonic functions, notion of weak solutions of an elliptic PDE);
- definition of deRham's cohomology and deRham's Theorem;
- covering spaces.
Teaching methods
Taught class, assigned exercises in class to be tackled in groups, assigned homework to be corrected in class in an interactive mode.
Teaching Resources
- P. Petersen, "Riemannian Geometry" (3rd ed.). Grad. Texts in Math. 171, Springer, Cham, 2016, xviii+499 pp.
- M.P. Do Carmo, "Riemannian Geometry", Math. Theory Appl. Birkhäuser Boston, Inc., Boston, MA, 1992, xiv+300 pp.
- I. Chavel, "Riemannian geometry—a modern introduction", Cambridge Tracts in Math., 108, Cambridge University Press, Cambridge, 1993, xii+386 pp.
- M. Dajczer and R. Tojeiro, "Submanifold theory", Universitext, Springer, New York, 2019, xx+628 pp.
Other references indicated by the teacher.
- M.P. Do Carmo, "Riemannian Geometry", Math. Theory Appl. Birkhäuser Boston, Inc., Boston, MA, 1992, xiv+300 pp.
- I. Chavel, "Riemannian geometry—a modern introduction", Cambridge Tracts in Math., 108, Cambridge University Press, Cambridge, 1993, xii+386 pp.
- M. Dajczer and R. Tojeiro, "Submanifold theory", Universitext, Springer, New York, 2019, xx+628 pp.
Other references indicated by the teacher.
Assessment methods and Criteria
Oral exam on the topics of the course, whose structure will be agreed with the teacher.
MATH-02/B - Geometry - University credits: 6
Lessons: 42 hours
Professors:
Colombo Giulio, Mari Luciano
Professor(s)
Reception:
by appointment (contact me by e-mail)
room 2060 (2nd floor) - Department of Mathematics - via Cesare Saldini 50, Milano
Reception:
Please contact me via email to fix an appointment
Math Department "Federigo Enriques"