Fischer–Marsden Conjecture on Hypersurfaces in the Complex Projective Space

By using the notion of Fischer–Marsden equation on real hypersurfaces in the complex projective space $\mathbb{C} P^{n} = SU_{n+1}/S(U_{1} U_{n})$, we can assert that there does not exist a nontrivial solution $(g,\nu)$ of such equation if the real hypersurface has isometric Reeb flow. Next, as an application we also show that there does not exist a nontrivial solution $(g,\nu)$ of the Fischer–Marsden equation on contact real hypersurfaces except when the radius $r_{1} \in (0,r_{0})$ for a tube over a real projective space $\mathbb{R} P^{n}$ in the complex projective space $\mathbb{C} P^{n}$. Consequently, the Fischer–Marsden conjecture is true on these two kinds of real hypersurfaces in the complex projective space $\mathbb{C} P^{n}$.