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.