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New analytical expressions are found for the Levi-Civita symbol using the Kronecker delta symbol. The expressions are derived up to 3 dimensions, extended to higher dimensions, and confirmed in Matlab for 5 dimensions. The expressions can be re-cast in terms of elementary and/or special functions, which lead to the conclusion that the Levi-Civita Symbol can be treated as a generalized, discrete function
We prove the existence and uniqueness of Levi-Civita connections for strongly sigma-compatible pseudo-Riemannian metrics on tame differential calculi. Such pseudo-Riemannian metrics properly contain the classes of bilinear metrics as well as their co
We give a new definition of Levi-Civita connection for a noncommutative pseudo-Riemannian metric on a noncommutative manifold given by a spectral triple. We prove the existence-uniqueness result for a class of modules of one forms over a large class
A concise analytical formula is developed for the inverse of an invertible 3 x 3 matrix using a telescoping method, and is generalized to larger square matrices. The formula is confirmed using randomly generated matrices in Matlab
Applying the Pomeransky inverse scattering method to the four-dimensional vacuum Einstein equation and using the Levi-Civita solution for a seed, we construct a cylindrically symmetric single-soliton solution. Although the Levi-Civita spacetime gener
We study covariant derivatives on a class of centered bimodules $mathcal{E}$ over an algebra A. We begin by identifying a $mathbb{Z} ( A ) $-submodule $ mathcal{X} ( A ) $ which can be viewed as the analogue of vector fields in this context; $ mathca