Generalized Curvature and Torsion in Nonstandard Analysis: Nonstandard Technical Treatment for Some Differential Geometry Concepts

The two notions of curvature and torsion have a great role in differential geometry and they have more applications in different scientific areas like: computer vision image improvement, dielectric surfaces, chemical interatomic surfaces, biological membranes, and double bubbles). There are several classical studies of curvature and some nonstandard studies about differential geometry, but none of them treated the problem of curvature at or near singular points. Black holes are one of a real examples about singularity which has highly energy at its singular point that make hastily change of the value of the curvature of any curve pass throughout it. Our investigation throughout this book is to use nonstandard tools for treating such problems, the study of this subject leads to introduce a more general definition about curvature and torsion which enable us to find the curvature and torsion values of points includes in an infinitesimal neighborhood of singularities rather than, finding the value of it at ordinary points.