In this talk, we investigate a pseudo-Ricci-Bourguignon soliton on real hypersurfaces in the complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$. By using pseudo-anti commuting Ricci tensor, we give a complete classification of Hopf pseudo-Ricci-Bourguignon soliton real hypersurfaces in $G_2({\mathbb C}^{m+2})$ . Moreover, we have proved that there exists a non-trivial classification of gradient pseudo-Ricci-Bourguignon soliton $(M, {\xi}, {\eta}, {\Omega}, {\theta}, {\gamma}, g)$ on real hypersurfaces with isometric Reeb flow in the complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$. In the class of contact hypersurface in $G_2({\mathbb C}^{m+2})$, we prove that there does not exist a gradient pseudo-Ricci-Bourguignon soliton in $G_2({\mathbb C}^{m+2})$

In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field \(T\), that is, \(R_{\xi}\phi T=TR_{\xi}\phi\), where \(T=A\) or \(T=S\) for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators, We give a complete classification of real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting condition respectively.

There are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians $G_{2}(\mathbb{C}^{m+2})$ Among them, Suh classified a Hopf hypersurface $M$ in $G_{2}(\mathbb{C}^{m+2})$ with Reeb parallel Ricci tensor in Levi-Civita connection. In this paper, we introduce a new notion of generalized Tanaka-Webster Reeb parallel Ricci tensor for $M$ in $G_{2}(\mathbb{C}^{m+2})$. Related to such a notion, we give some characterizations for a real hypersurface of Type~$(A)$ in $G_{2}(\mathbb{C}^{m+2})$.