Lie groupoids and lie algebroids in differential geometry

Tangent and cotangent bundles : differential geometry

Differential geometry on complex and almost complex spaces

Structures on Manifolds

Semi-Inner Products, that can be naturally defined in general Banach spaces over the real or complex number field, play an important role in describing the geometric properties of these spaces.In the first chapter of the book, a short introduction to the main properties of the duality mapping that will be used in the next chapters is given. Chapter 2 is devoted to the semi-inner products in the sense of Lumer-Giles while the 3rd chapter is concerning with the main properties of the superior and inferior semi-inner products. In the next chapter the main properties of Milicics semi-inner product and the properties of normed spaces of () — type are presented. The next two chapters investigate the geometric properties of (), ()and 2–inner product spaces introduced by the author, while Chapter 7 is entirely devoted to the study of different mappings that can naturally be associated to the norm derivatives in general normed spaces and, in particular, in inner product spaces. Chapters 8 and 9 investigate different orthogonalities that may be introduced in normed spaces and their intimate relationship with semi-inner products. In Chapter 11, orthogonal decomposition theorems in general normed spaces are provided, while in the next chapter the problem of approximating continuous linear functionals in general normed spaces and characterizations of reflexivity in this context are given. A deeper insight on this problem is then considered in Chapter 13, where some classes of continuous functionals are introduced and a density result based on the famous Bishop-Phelps theorem is obtained. In Chapter 14, the class of smooth normed spaces of (BD)-type and their application for non-linear operators is presented. In the next chapter the continuous sublinear functionals defined in Reflexive Banach spaces is investigated, while Chapter 16 deals with convex functions defined in more general spaces endowed with subinner products. The monograph concludes by considering the representation problem of linear forms defined on modules endowed with general semi-subinner products.

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Semi-Inner Products and Applications

A research on the properties of the Golden structure (i.e. a polynomial structure with the structure polynomial Q ( X ) = X 2 - X - I ) is carried out in this article. The Golden proportion plays a central role in this paper. The geometry of the Golden structure on a manifold is investigated by using a corresponding almost product structure.

https://www.math.uaic.ro/~mcrasm/depozit/l47.pdf

Golden differential geometry

The aim of this paper is to study two problems in the framework of paracontact geometry of dimension $3$, namely, the class of parallel symmetric tensor fields of $(0, 2)$-type and possible Lorentz Ricci solitons. We search for two types of Ricci solitons: the first when its potential vector field is exactly the characteristic vector field $\xi $ of the paracontact structure and the second when the potential vector field is a paracontact-holomorphic one. For the former case we find all variants of Ricci solitons, expanding, steady and shrinking, and the fact that $\xi $ is a conformal Killing vector field. A class of examples is completely discussed.

https://www.math.uaic.ro/~mcrasm/depozit/l85.pdf

Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry

Our aim in this paper is to focus on some applications in differential geometry of the metallic means family (a generalization of the golden mean) and generalized Fibonacci sequences, using a class of polynomial structures defined on Riemannian manifolds. We search for properties of the induced structure on a submanifold by metallic Riemannian structures and we find a necessary and sufficient condition for a submanifold to be also a metallic Riemannian manifold in terms of invariance. Also, the totally geodesic, minimal and respectively totally umbilical hypersurfaces in metallic Riemannian manifolds are analyzed and the Euclidean space and its hypersphere is treated as example.

Metallic structures on Riemannian manifolds

The Eisenhart problem of finding parallel tensors is solved for the symmetric case in the regular f-Kenmotsu framework. In this way, the Olszack-Rosca example of Einstein manifolds provided by f-Kenmotsu manifolds via locally symmetric Ricci tensors is recovered as well as a case of Killing vector fields. Some other classes of Einstein-Kenmotsu manifolds are presented. Our result is interpreted in terms of Ricci solitons and special quadratic first integrals.

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From the Eisenhart problem to Ricci solitons in $f$-Kenmotsu manifolds

The Eisenhart problem of finding parallel tensors treated already in the framework of quasi-constant curvature manifolds in \cite{x:j} is reconsidered for the symmetric case and the result is interpreted in terms of Ricci solitons. If the generator of the manifold provides a Ricci soliton then this is i) expanding on para-Sasakian spaces with constant scalar curvature and vanishing $D$-concircular tensor field and ii) shrinking on a class of orientable quasi-umbilical hypersurfaces of a real projective space=elliptic space form.

Mircea Crasmareanu

Papers

Golden differential geometry

Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry

Metallic structures on Riemannian manifolds

From the Eisenhart problem to Ricci solitons in $f$-Kenmotsu manifolds

Ricci solitons in manifolds with quasi-constant curvature