Lee, J.M. (2018) Introduction to Riemannian Manifolds. 2nd Edition, Springer.
https://doi.org/10.1007/978-3-319-91755-9

has been cited by the following article:

• TITLE: An Angular Reformulation of the Kronecker Delta through a Jeno-Type Structural Nullity—Gram-Schmidt Orthogonalization and a Sliding-Ladder Application

  AUTHORS: Oscar Mauricio Jeno Soto

  KEYWORDS: Kronecker Delta, Gram Matrix, Gram-Schmidt Orthogonalization, Sliding Ladder, Unilateral Contact, Orthogonality, Structural Nullity

  JOURNAL NAME: Journal of Applied Mathematics and Physics, Vol.14 No.8, August 24, 2026

  ABSTRACT: A geometric reformulation of the Kronecker delta is proposed for finite-dimensional real inner-product spaces equipped with normalized basis vectors. The construction separates index identity, represented by δ ij , from off-diagonal angular coupling, represented by ( 1− δ ij )cos θ ij . The resulting quantity coincides with the Gram matrix for normalized bases and reduces exactly to the classical Kronecker delta for orthonormal bases. The label structural nullity distinguishes an off-diagonal pair whose metric coupling vanishes by orthogonality; it does not introduce a new arithmetic zero. Two applications are developed. First, Gram-Schmidt orthogonalization is interpreted as the systematic elimination of off-diagonal Gram couplings, quantified by the sign-independent functional N( B )= ∑ i