Differential Geometry and Matrix-Based Generalizations of the Pythagorean Theorem in Space Forms

In this work, we consider Pythagorean triples and quadruples using fundamental form matrices of hypersurfaces in three- and four-dimensional space forms and illustrate various figures. Moreover, we generalize that an immersed hypersphere Mn with radius r in an (n+1)-dimensional Riemannian space form Mn+1(c), where the constant sectional curvature is c∈{−1,0,1}, satisfies the (n+1)-tuple Pythagorean formula Pn+1. Remarkably, as the dimension n→∞ and the fundamental form N→∞, we reveal that the radius of the hypersphere converges to r→12. Finally, we propose that the determinant of the Pn+1 formula characterizes an umbilical round hypersphere satisfying k1=k2=⋯=kn, i.e., Hn=Ke in Mn+1(c).