Riemannian Geometry
A.Y. 2025/2026
Learning objectives
The aim of the course i sto introduce the student to some advanced topic in the classical geometry of surfaces in Euclidean space
Expected learning outcomes
A working knowledge of the moving frame and of the analytical tools in the study of differential geometry.
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
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)
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)
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 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.
- (for a proof of Hodge's Theorem) S. Rosenberg, "The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds", London Mathematical Society Student Texts, Series Number 31, 1st Edition.
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.
- (for a proof of Hodge's Theorem) S. Rosenberg, "The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds", London Mathematical Society Student Texts, Series Number 31, 1st Edition.
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.
Professor(s)
Reception:
Please contact me via email to fix an appointment
Math Department "Federigo Enriques"