Array Manifold Curves in ${Fraktur {C}}^{N}$ and Their Complex Cartan Matrix

The differential geometry of array manifold curves has been investigated extensively in the literature, leading to numerous applications. However, the existing differential geometric framework restricts the Cartan matrix to be purely real and so the vectors of the moving frame BBU(s) are found to be orthogonal only in the wide sense (i.e. only the real part of their inner product is equal to zero). Imaginary components are then accounted for separately using the concept of the inclination angle of the manifold. The purpose of this paper is therefore to present an alternative theoretical framework which allows the manifold curve in FrakturC^N to be characterized in a more convenient and direct manner. A continuously differentiable strictly orthonormal basis is established and forms a platform for deriving a generalized complex Cartan matrix with similar properties to those established under the previous framework. Concepts such as the radius of circular approximation, the manifold curve radii vector and the frame matrix are also revisited and rederived under this new framework.