Shape Operators of a Directional Tubular Surface in 4-Space

This paper examines a tubular surface, a specific example of a canal surface, in 4-dimensional Euclidean space. In the plane stretched by the quasi-frame vectors Bq and Cq, this surface is established by the motion of a circle with a constant radius that uses each point on the curve a (t) as its center. Using the general equation provided in Euclidean 4-space, the first and second partial derivatives are determined. The Gram-Schmidt technique was used to derive the surface's first unit normal vector field U₁, and second unit normal vector field U₂, using the acquired partial derivatives. Using quasi-vectors, the tubular surface's first and second fundamental form coefficients were found. Furthermore, the shape operator matrices for the tubular surface's the unit normal vector fields U₁ and U₂ were acquired. We have found algebraic invariants of the shape operator, Gaussian curvature, and mean curvature. For a thorough understanding of the obtained theoretical calculations, an example of a directional tubular surface, the equation of the tubular surface has been parametrized using quasi-frame vectors and quasi-frame curvatures for a given space curve in 4-dimensional Euclidean space.