Multiplicative Differential Geometry

This book introduces multiplicative Frenet curves. We define multiplicative tangent, multiplicative normal, and multiplicative normal plane for a multiplicative Frenet curve. We investigate the local behaviours of a multiplicative parameterized curve around multiplicative biregular points, define multiplicative Bertrand curves and investigate some of their properties. A multiplicative rigid motion is introduced.

The book is addressed to instructors and graduate students, and also specialists in geometry, mathematical physics, differential equations, engineering, and specialists in applied sciences. The book is suitable as a textbook for graduate and under-graduate level courses in geometry and analysis. Many examples and problems are included.

The author introduces the main conceptions for multiplicative surfaces: multiplicative first fundamental form, the main multiplicative rules for differentiations on multiplicative surfaces, and the main multiplicative regularity conditions for multiplicative surfaces. An investigation of the main classes of multiplicative surfaces and second fundamental forms for multiplicative surfaces is also employed. Multiplicative differential forms and their properties, multiplicative manifolds, multiplicative Einstein manifolds and their properties, are investigated as well.

Many unique applications in mathematical physics, classical geometry, economic theory, and theory of time scale calculus are offered.

本書介紹了乘法 Frenet 曲線。我們為乘法 Frenet 曲線定義了乘法切線、乘法法線和乘法法平面。我們研究了乘法參數化曲線在乘法雙正則點附近的局部行為，定義了乘法 Bertrand 曲線並研究了它們的一些性質。我們引入了乘法剛體運動。

本書針對教師、研究生和幾何學、數學物理學、微分方程、工程學以及應用科學專家進行了撰寫。本書適合作為研究生和本科水平的幾何學和分析課程的教材。書中包含了許多例子和問題。

作者介紹了乘法曲面的主要概念：乘法第一基本形式、乘法曲面上的微分法則以及乘法曲面的主要正則條件。同時還對乘法曲面的主要類別和乘法曲面的第二基本形式進行了研究。還研究了乘法微分形式及其性質、乘法流形、乘法愛因斯坦流形及其性質。

本書還提供了在數學物理學、經典幾何學、經濟理論和時間尺度微積分理論中的許多獨特應用。

Svetlin G. Georgiev is a mathematician who has worked in various areas of the study. He currently focuses on harmonic analysis, functional analysis, partial differential equations, ordinary differential equations, Clifford and quaternion analysis, integral equations, and dynamic calculus on time scales. He is also the author of Dynamic Geometry of Time Scales, CRC Press. He is a co-author of Conformable Dynamic Equations on Time Scales, with Douglas R. Anderson, and also: Multiple Fixed-Point Theorems and Applications in the Theory of ODEs, FDEs and PDE; Boundary Value Problems on Time Scales, Volume 1 and Volume II, all with Khalid Zennir and published by CRC Press.

Svetlin G. Georgiev是一位數學家，他在研究的各個領域有所貢獻。他目前專注於調和分析、泛函分析、偏微分方程、常微分方程、Clifford和四元數分析、積分方程和時間尺度上的動態微積分。他也是CRC Press出版的《時間尺度的動態幾何》的作者。他與Douglas R. Anderson合著了《時間尺度上的可調動態方程》。此外，他還與Khalid Zennir合著了《ODEs、FDEs和PDE理論中的多重不動點定理及應用》和《時間尺度上的邊界值問題，第一卷》和《第二卷》，這些書籍也由CRC Press出版。