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

Erhan Güler (), Yusuf Yaylı and Magdalena Toda
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Erhan Güler: Department of Mathematics, Bartın University, 74100 Bartın, Türkiye
Yusuf Yaylı: Department of Mathematics, Ankara University, 06100 Ankara, Türkiye
Magdalena Toda: Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA

Mathematics, 2025, vol. 13, issue 5, 1-21

Abstract: 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 ) .

Keywords: space forms; Pythagorean triples; Pythagorean quadruples; Pythagorean ( n + 1)-tuples; hypersurface; hypersphere; radius; fundamental form matrices (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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