Differential Geometry Lecture 6 (Part 1): Fundamental Theorem of Space curves; Frenent approximation

This course on Differential Geometry is intended for science majors who need to have knowledge about the geometry of curves and surfaces in space and want to have an introduction to manifolds. It is especially appropriate for math, physics and astronomy majors. @RUeamHK0X6#

[quantumofspace]

There is a great idea! For the dark side of the Universe - suppose that it consists of short-term interactions in long-lived fractal networks, the smallest quantum operators in energy, spherical rosebuds, consisting of a large set; 1 - rolled into a sphere, 2 - half collapsed into a sphere and 3 - flat, vibrating quantum membranes relative to their working centers in the sphere.

[muzzamilhussain]

Sir please camera Sahi nahi hota ap jis ki waja sa smaj nahi ata sir Kya samjaraha

[AsgharAli-pqlt]

You have misstated the fundamental theorem of space curves. Two curves with the same nowhere vanishing curvature and torsion are related by a direct (or orientation preserving) isometry. Direct isometries are just rigid motions. Direct isometries/ rigid motions are compositions of translations and rotationtions, and exclude reflections but you say that two curves with the same curvature and torsion could be related by a translation and a reflection as well. Reflections are isometries of a Euclidean space but such isometries are excluded from the fundamental theorem of space curves. Your statement violates the fundamental theorem of space curves.


Further, your justification for the condition of nonzero curvature in the fundamental theorem of space curves is also incorrect.You say zero curvature results in non-interesting geometry. The reason that curves with zero curvatures are excluded from the fundamental theorem of space curve is that the proof of this theorem relies on Ferenet-Serret equations, and these equations hold only for the curves with nonzero curvature. This follows because if the curvature of a curve is zero at a point, then the principal normal vector of the curve is not defined at that point, and the Frenet-Serret equation no longer hold.


Also, I believe that the knowledge of the theory of space curves and surfaces although helpful in understanding the theory of manifolds (be it differential topology or differential geometry of manifolds), is not necessary to study a course on manifolds contrary to what is written in the introduction of this course.