Axial Presentations of Regular Arcs on M(n)

Area of Science:

• Differential Geometry
• Topology

Background:

• Riemannian manifolds are fundamental in geometry, providing a framework to study curved spaces.
• Understanding the local behavior of curves within these manifolds is crucial for various mathematical and physical applications.

Purpose of the Study:

• To establish a coordinate system that simplifies the representation of curves in Riemannian manifolds.
• To prove the existence of a local embedding for regular curves into a Euclidean domain.

Main Methods:

• The study utilizes the properties of Riemannian manifolds and the concept of algebraic arc length.
• It involves constructing a local presentation (F: U, X) of the manifold.

Main Results:

• A theorem is presented proving the existence of a local coordinate system.
• This system maps a simple, regular curve 'g' onto a segment of the x(1)-axis in a Euclidean domain.
• Each point p(s) on the curve is represented as (s,0,...,0).

Conclusions:

• The existence of such a local representation is confirmed for any smooth curve on a Riemannian manifold.
• This provides a simplified, standard local view of curves within complex manifold structures.