### Objectives

General targets

to get basic knowledge in riemannian geometry and more specifically in holonomy theory

Specific targets

Knowledge and understanding:
at the end of this teaching unit, the student shall have the basic notions and results about riemannian manifolds, vector bundles, the different notions of curvature, holonomy theory and riemannian holonomy, with particular attention to irreducible holonomy groups (Kähler manifolds, Calabi-Yau, hyperKähler, etc…).

Apply knowledge and understanding
at the end of this teaching unit, the student shall be able to study advanced topics in riemannian, complex and Kähler geometry, as well as to solve complex problems in these areas.

Critical Judgemental skills
the student shall have the basics to analyse the analogies and relationships between the topics learnt and several arguments coming from differential and algebraic topology, algebraic and complex geometry.

Communication skills
ability of expose the contents of the teaching unit in the possible oral exam as well as in the theoretical questions in the written exam.

Learning ability
the gained knowledge shall permit the student to address a possible master thesis on advanced topics in riemannian, complex or Kähler geometry.

### SIMONE DIVERIO   Teacher profile

#### Programme

Differentiable manifolds, partitions of unity, riemannian metrics (definition and existence), isometries, curves length, examples.
Vector bundles, functorial constructions, volume form, riemannian volume element, integration, orientability, riemannian submanifolds, sphere and hyperbolic space. Product metric, riemannian coverings, flat tori, classification of two dimensional flat tori modulo isometries and homotheties.
Linear connection on vector bundles, matrix connection, gauge transformations, curvature, covariant derivative, induced connections, Bianchi identity.
Pull-back of a vector bundle, parallel transport, holonomy and restricted holonomy, parallel sections, metrics on a vector bundle, compatibility of a connection with the metric, torsion.
Compatibility with the metric and parallel transport, holonomy and orthogonal group, existence and uniqueness of the Levi-Civita connection.
Crash course in several complex variables: definition of holomorphic function, Cauchy formula, analyticity, analytic continuation, maximum principle, rank theorem, Hartogs' theorem.
Quasi-complex structure on a even dimensional real vector space, complexification, complexified of the dual vector space and dual vector space of the complexified vector space, (1,0) and (0,1) decomposition, bigrading on the exterior algebra, J-invariant scalar products nd hermitian products, fundamental form of a J-invariant scalar product.
Complex manifolds, quasi-complex structures, integrability, holomorphic tangent space, complexification, dual and decomposition, complex differential calculus, Dolbeault cohomology.
(1,0) and (0,1) connections, holomorphic vector bundles, canonical (0,1)-connection on a holomorphic vector bundle, its Dolbeault cohomology, Chern connection, hermitian manifolds, hermitian volume form.
Kähler metrics, cohomological necessary condition for the existence of Kähler metrics, relationship between Levi-Civita and theories for Kähler manifolds, holonomic interpretation of Kähler manifolds.
Ricci curvature, further symmetries of Riemann curvature tensor, first Bianchi identity, symmetry of the Ricci tensor, scalar curvature, Einstein manifolds.
Chern-Ricci curvature, Chern-Ricci form, and their equivalence with Ricci curvature in the Kähler case.
Holonomic interpretation of Ricci-flatness: restricted holonomy contained in SU(n), homotopic invariance of parallel transport for flat connections, first Chern class, debar lemma, statement of Aubin-Yau's theorem, existence of Kähler metrics with prescribed Ricci curvature, canonical bundle holomorphically trivial implies holonomy contained in SU(n).
Quaternion algebra, norm, conjugation, the group Sp(1) of unitary quaternions, quaternionic general linear group, standard hermitian form and compact symplectic group.
Holonomic interpretation of the compact symplectic group, hyperKähler manifolds, K3 surfaces.
Irreducibility, completeness, totally geodesic submanifolds, statement and scheme of the proof of the de Rham decomposition theorem. Definition of locally symmetric spaces, riemannian characterization, statement of Berger's classification theorem, K3 surfaces again.

S. Gallot, D. Hulin, J. Lafontaine, "Riemannian Geometry"
D. Huybrechts, "Complex Geometry"

#### Bibliography

S. Kobayashi, K. Numizu, "Foundations of Differential Geometry" J.-P. Demailly, "Complex Analytic and Differential Geometry" M. Gorss, D. Huybrechts, D. Joyce, "Calabi-Yau Manifolds and Related Geometries"

#### Prerequisites

Basic differential geometry, general topology, covering spaces and fundamental group, complex variable.

#### Exam modes

Written exam and possibly oral exam.

Exam reservation date start Exam reservation date end Exam date
22/10/2018 18/01/2019 23/01/2019
22/10/2018 08/02/2019 14/02/2019
22/10/2018 17/06/2019 21/06/2019
22/10/2018 12/07/2019 17/07/2019
22/10/2018 13/09/2019 19/09/2019
11/11/2019 24/11/2019 25/11/2019
Course sheet