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

Güler E., YAYLI Y., Toda M.

Mathematics, cilt.13, sa.5, 2025 (SCI-Expanded) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 13 Sayı: 5
  • Basım Tarihi: 2025
  • Doi Numarası: 10.3390/math13050836
  • Dergi Adı: Mathematics
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, Metadex, zbMATH, Directory of Open Access Journals, Civil Engineering Abstracts
  • Anahtar Kelimeler: fundamental form matrices, hypersphere, hypersurface, Pythagorean (n + 1)-tuples, Pythagorean quadruples, Pythagorean triples, radius, space forms
  • Ankara Üniversitesi Adresli: Evet

Özet

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 (Formula presented.) with radius r in an (Formula presented.) -dimensional Riemannian space form (Formula presented.), where the constant sectional curvature is (Formula presented.), satisfies the (Formula presented.) -tuple Pythagorean formula (Formula presented.). Remarkably, as the dimension (Formula presented.) and the fundamental form (Formula presented.), we reveal that the radius of the hypersphere converges to (Formula presented.). Finally, we propose that the determinant of the (Formula presented.) formula characterizes an umbilical round hypersphere satisfying (Formula presented.), i.e., (Formula presented.) in (Formula presented.).