DCE-Geometry: A Unified Framework for Dimensional and Curvature Extension

Objective: This paper develops a new geometric theory called DCE-Geometry, where D stands for Dimension, C stands for Curvature, and E stands for Extension. The theory simultaneously extends both the dimension parameter and the curvature parameter to complex numbers through analytic continuation, establishing a unified framework for Euclidean geometry (K = 0, n ∈ N), spherical geometry (K > 0, n ∈ N), hyperbolic geometry (K < 0, n ∈ N), and Riemannian geometry (variable K, n ∈ N). The fundamental object of study is the two-parameter constant curvature space MαK, where α ∈ C and K ∈ C.Innovation: 1. Dimensional Extension in DCE-Geometry: By introducing the negative-one-dimensional geometric unit N(−1) = 1, we establish the geometric characteristic expression NX+1k=0(−1)k N(N−k) = 0 and the total sum expression NX+1k=0 N(N−k) = 2N+1, proving their one-to-one correspondence with the expansion coefficients of (a ± b)N+1. 2. Curvature Extension in DCE-Geometry: Extending the curvature parameter K to complex numbers, we establish the curvature-parameterized unified law of cosines: CosK(c) = CosK(a) CosK(b) + SinK(a) SinK(b) cos C, where CosK(x) = cos(√Kx), SinK(x) = √1Ksin(√Kx). 3. Two-Parameter Unification: Simultaneously extending dimension α and curvature K to complex numbers, we construct the two-parameter constant curvature space MαK, proving that its metric, curvature, volume, GaussBonnet integral, Riemann-Roch integral, and index are all analytic functions of (α, K). 4. Extension of Core Theorems: The Gauss-Bonnet theorem, Riemann-Roch theorem, and Atiyah-Singer index theorem are extended to DCE-geometry, with explicit degenerate cases for Euclidean, spherical, hyperbolic, and complex curvature geometries.Methods: Using differential geometry, complex analysis, and analytic continuation, we construct real-dimensional and complex-dimensional constant curvature spaces and Riemannian manifolds through Gamma functions and formal power series. The dimensional extension axiomatic system from references [1,2] is systematically applied to geometric quantities in real and complex dimensions.Main Results: 1. Dimensional Extension in DCE-Geometry: Rigorous construction of real-dimensional Euclidean spaces Er, spherical spaces Sr, hyperbolic spaces Hr(r ∈ R) and their complex-dimensional counterparts (α ∈ C), including metric construction, distance definitions, Pythagorean theorem, law of sines, law of cosines, and volume formulas. 2. Curvature Extension in DCE-Geometry: Extension of curvature K to complex numbers, proving the unified metric ds2 = Pdx2i.