Ferreira, MiltonSuksumran, Teerapong2020-07-152020-07-152020-06-03Ferreira, M.; Suksumran, T. Orthogonal Gyrodecompositions of Real Inner Product Gyrogroups. Symmetry 2020, 12(6), 941.http://hdl.handle.net/10400.8/5008In this article, we prove an orthogonal decomposition theorem for real inner product gyrogroups, which unify some well-known gyrogroups in the literature: Einstein, M\"{o}bius, Proper Velocity, and Chen's gyrogroups. This leads to the study of left (right) coset partition of a real inner product gyrogroup induced from a subgyrogroup that is a finite dimensional subspace. As~a result, we obtain gyroprojectors onto the subgyrogroup and its orthogonal complement. We~construct also quotient spaces and prove an associated isomorphism theorem. The left (right) cosets are characterized using gyrolines (cogyrolines) together with automorphisms of the subgyrogroup. With~the algebraic structure of the decompositions, we study fiber bundles and sections inherited by the gyroprojectors. Finally, the general theory is exemplified for the aforementioned gyrogroups.engReal inner product gyrogroupOrthogonal decompositionGyroprojectionCoset spacePartitionsQuotient spaceGyrolinesCogyrolinesfiber bundlesjournal article10.3390/sym12060941