دانلود کتاب نابرابری‌های هندسی و کاربردها

Geometric inequalities play a crucial role in various branches of science and engineering, providing foundational tools for theoretical development and problem-solving. In mathematics, applications of geometric inequalities span across geometry, analysis, and optimization, among many others. For instance, the priori estimates for differential equations and the theory of functions of complex variables provide many such examples. In physics, particularly in general relativity and string theory, geometric inequalities help describe spacetime geometries and their properties under various physical conditions. For example, physicists can use geometric inequalities to derive constraints on energy distributions based on geometric configurations. Geometric inequalities also play a crucial role in geometric analysis, particularly in studying elliptic partial differential equations related to geometric _lows. They help establish regularity results for solutions to these equations by controlling various norms associated with functions defined on (Riemannian) manifolds. Also, the Sobolev inequalities provide bounds on functions defined on manifolds concerning their derivatives and integrals over certain domains. These inequalities are instrumental in studying partial differential equations on manifolds and have implications for embedding theorems. In differential geometry, geometric inequalities play a crucial role, providing important tools for understanding the properties of geometric objects, such as Riemannian manifolds and their submanifolds. In addition, geometric inequalities help to analyze curvature properties of manifolds. For example, the Cauchy-Schwarz inequality has been applied to derive bounds on sectional curvature, which is crucial in studying Riemannian manifolds with constant curvature, such as spheres and hyperbolic spaces. Moreover, the relationship between curvature and geometric inequalities allows differential geometers to classify manifolds based on their curvature characteristics. In addition, geometric inequalities facilitate comparison theorems that relate different geometric structures. For example, Bonnet-Myers’ theorem uses geometric inequalities to prove that if a Riemannian manifold has positive Ricci curvature, then it is always compact and it has a finite diameter. Such results are pivotal in understanding the global structure of Riemannian manifolds. In addition, embedding theorems often rely on geometric inequalities to establish conditions under which a manifold can be embedded into Euclidean space. For instance, the Whitney and Nash embedding theorems utilize concepts from geometric inequalities to demonstrate that any smooth manifold can be embedded into a Euclidean space of higher dimension. This book is devoted to recent advances in a variety of geometric inequalities in differential geometry, as well as in the theory of solitons. As a result, this book consists of 15 chapters authored by leading mathematicians, encompassing a wide array of topics, including “Some Inequalities for Geometric Solitons” by A. M. Blaga, “Generalized Ricci- Yamabe Soliton on 3-Dimensional Lie Groups” by A. Delloum and G. Beldjilali, “Riemannian Invariants in Submanifold Theory” by A. Mihai, “Chen Inequalities for Submanifolds of Kenmotsu Space Forms” by I. U􀇆 nal, A. Barman and D. G. Prakasha, “Improved Chen-Ricci Inequalities for Semi-slant ξ⊥-Riemannian Submersions from Sasakian Space Forms” by M. A. Akyol and N. Poyraz, “Characterizations of Perfect Fluid and Generalized Robertson-Walker Space-Time Admitting k Almost Ricci Yamabe Solitons” by K. De and U. C. De, “Riemannian Concircular Structure Manifold and Solitons” by S. K. Chaubey and A. Haseeb, “Statistical Maps and a Chen’s First Inequality for These Maps” by S. Kazan and A. N. Suddiqui, “Hyperbolic Ricci-Yamabe Solitons and η- Hyperbolic Ricci-Yamabe Solitons” by M. D. Siddiqi, “A Survey on Hitchin-Thorpe Inequality and Its Extensions” by B.-Y. Chen, M. A. Choudhary, and M. Nisar, “The Principal Eigenvalue of a (p, q)- Biharmonic System Along the Ricci Flow” by S. Azami and Gh. Fasihi- Ramandi, “The Jacobi Geometry of Plane, Parametrized Curves and Associated Inequalities” by M. Crasmareanu, “B.-Y. Chen Inequalities for Submanifolds of a Conformally Flat Manifold” by C. O􀇆 zür, “General Chen Inequalities for Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature” by S. Decu and G.-E. Vilcu, and “B. Y. Chen Inequalities for Pointwise Quasi Hemi-Slant Submanifolds of a Kaehler Manifold” by N. Poyraz, M. A. Akyol, and Erol Yasar. Both editors of this book hope that readers will find this book a valuable reference for geometrical inequalities, enabling them to perform their research more effectively, successfully, and creatively.