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 M n with radius r in an ( n + 1 ) -dimensional Riemannian space form M n + 1 ( c ) , where the constant sectional curvature is c ∈ { − 1 , 0 , 1 } , satisfies the ( n + 1 ) -tuple Pythagorean formula P n + 1 . Remarkably, as the dimension n → ∞ and the fundamental form N → ∞ , we reveal that the radius of the hypersphere converges to r → 1 2 . Finally, we propose that the determinant of the P n + 1 formula characterizes an umbilical round hypersphere satisfying k 1 = k 2 = ⋯ = k n , i.e., H n = K e in M n + 1 ( c ) .