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.