Fischer–Marsden Conjecture and Equation in the Complex Hyperbolic Quadric

Abstract

By using the notion of Fischer–Marsden equation on real hypersurfaces in the complex hyperbolic quadric ${{Q^m}^*} = {{\text {SO}}^o_{2,m}/{\text {SO}}_2 {\text {SO}}_m}$, we can assert that there does not exist a non-trivial solution $(g,{\nu })$ of Fischer–Marsden equation on real hypersurfaces with isometric Reeb flow in the complex hyperbolic quadric ${Q^m}^*$. Next, as an application we also show that there does not exist a non-trivial solution $(g,{\nu })$ of the Fischer–Marsden equation on contact real hypersurfaces in the complex hyperbolic quadric ${Q^m}^*$. Consequently, the Fischer–Marsden conjecture is true on these two kinds of real hypersurfaces in the complex hyperbolic quadric ${Q^m}^*$.

Keywords

Fischer–Marsden equation / $(g, \nu )$')">Non-trivial solution $(g, \nu )$ / $\mathfrak {A}$-Isotropic')">$\mathfrak {A}$-Isotropic / $\mathfrak {A}$-Principal')">$\mathfrak {A}$-Principal / Complex conjugation / Complex hyperbolic quadric / 53C40 / 53C55

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Funding

National Research Foundation of Korea(NRF 2018-R1D1A1B-05040381)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature