**Preface to "Inequalities in Geometry and Applications"**

Geometric inequalities have fascinated the mathematical world and not only since ancient times, as this field of research is in fact as old as mathematics itself. Over time, such inequalities have proven to be an excellent tool in investigating and solving basic problems in pure and applied sciences, including some that were apparently unrelated to geometric inequalities.

The aim of this book was to present recent developments in the field of geometric inequalities and their applications. The volume covers a vast range of topics, such as isoperimetric problem, Erdos–Mordell inequality, Barrow's inequality, Simpson inequality, Chen inequalities, q-integral ¨ inequalities, complex geometry, contact geometry, statistical manifolds, Riemannian submanifolds, optimization theory, topology of manifolds, log-concave functions, Obata differential equation, o-invariants, Einstein spaces, warped products, and solitons. By exposing new concepts, techniques and ideas, this book will certainly stimulate further research in this field.

Reviewed by leading experts, the chapters in this book were written by scientists from 13 different countries, most of them being outstanding researchers in the field. I am thankful to all the contributors and also to the journal *Mathematics* for giving me the opportunity to publish this book.

> **Gabriel-Eduard Vˆılcu** *Editor*

## *Article* **A Closed Form for Slant Submanifolds of Generalized Sasakian Space Forms**

#### **Pablo Alegre 1,\*, Joaquín Barrera 2 and Alfonso Carriazo 2,†**


Received: 4 November 2019; Accepted: 9 December 2019; Published: 13 December 2019

**Abstract:** The Maslov form is a closed form for a Lagrangian submanifold of C*<sup>m</sup>*, and it is a conformal form if and only if *M* satisfies the equality case of a natural inequality between the norm of the mean curvature and the scalar curvature, and it happens if and only if the second fundamental form satisfies a certain relation. In a previous paper we presented a natural inequality between the norm of the mean curvature and the scalar curvature of slant submanifolds of generalized Sasakian space forms, characterizing the equality case by certain expression of the second fundamental form. In this paper, first, we present an adapted form for slant submanifolds of a generalized Sasakian space form, similar to the Maslov form, that is always closed. And, in the equality case, we studied under which circumstances the given closed form is also conformal.

**Keywords:** slant submanifolds; generalized Sasakian space forms; closed form; conformal form; Maslov form
