It is well known that transformations of $ \mathbb{C}^n $ preserving the standard inner product are unitary transformations. In this paper, all bijective transformations of isotropic sets of $ \mathbb{C}P^{n} $ preserving $ H $-orthogonality in both directions, called $ H $-orthogonal transformations, have been determined. This is a generalization of Uhlhorn's version of Wigner's unitary-antiunitary theorem. The group of $ H $-orthogonal transformations on some other sets of $ \mathbb{C}P^{n} $ were also determined.
Citation: Kai Zhou, Zhenhua Gu, Hongfeng Wu. Orthogonality preserving transformations on complex projective spaces[J]. AIMS Mathematics, 2025, 10(5): 11411-11434. doi: 10.3934/math.2025519
It is well known that transformations of $ \mathbb{C}^n $ preserving the standard inner product are unitary transformations. In this paper, all bijective transformations of isotropic sets of $ \mathbb{C}P^{n} $ preserving $ H $-orthogonality in both directions, called $ H $-orthogonal transformations, have been determined. This is a generalization of Uhlhorn's version of Wigner's unitary-antiunitary theorem. The group of $ H $-orthogonal transformations on some other sets of $ \mathbb{C}P^{n} $ were also determined.
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Kai Zhou, Zhenhua Gu, Hongfeng Wu. Orthogonality preserving transformations on complex projective spaces[J]. AIMS Mathematics, 2025, 10(5): 11411-11434. doi: 10.3934/math.2025519
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