Supervisors info:
Λάππας Διονύσιος, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, Σχολή Θετικών Επιστημών
Ανδρουλιδάκης Ιάκωβος, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, Σχολή Θετικών Επιστημών
Γιαννιώτης Παναγιώτης, Επίκουρος Καθηγητής, Τμήμα Μαθηματικών, Σχολή Θετικών Επιστημών
Summary:
At this thesis, we are going to study basic elements of complex structures on manifolds.
In the first chapter, basic concepts of real manifolds are reminded, i.e. manifolds whose charts are maps to real vector spaces, like differential curves, the tangent space, the tangent bundle, the flow of a vector space and the differential. Subsequently, the definition of Riemannian manifolds are given, with its metric, the isometry of these manifolds and the Riemann connection, pointing out conditions for its uniqueness, which allows us to introduce different types of curvatures. At the end of this chapter, elements of tensors and differential forms are given, which are useful for the integration of manifolds.
In the second chapter, we give the definition of complex manifolds and the structures of them, giving some examples of manifolds with this form. The chapter begins with some attributes of the complex algebra, such as the endomorphism J with J^2=-1 and the complexification of a real vector space. The concept of holomorphic functions is introduced, that satisfy the equations Cauchy – Riemann, which is going to be used at the definition of tangent complex space. Subsequently, the definition of metrics on complex manifolds is given, such as the Hermitian and Kahler metric, concluding the curvature of Kahler manifolds.
Third chapter ‘s subject deals with holomorphic tangent bundles, emphasising on holomorphic line bundles and holomorphic tangent bundle. The Kodaira ’s embedding theorem is mentioned, according to which, through a positive line bundle, we have a holomorphic embedding of a complex manifold on a complex projective space.
The fourth and last chapter deals with the symplectic theory and the definition of symplectic manifolds. Starting with the basic elements of this theory, symplectic structures are connected with almost complex structures, ending in two cases of integration of almost complex structure.
Keywords:
Complex manifolds, Almost complex structure, Hermitian metric, Kahler metric, Holomorphic vector bundles, Symplectic theory
