In this paper, we consider the Hyperbolic Ricci-Bourguignon ow on a compact man-ifold M and show that this ow has a unique solution on short-time with imposing on initial conditions. After then, we , nd evolution equations for Riemannian cur-vature tensor, Ricci curvature tensor and scalar curvature of M under this ow. In the , nal section, we give some examples of this ow on some compact manifolds.

In this paper, we consider Hyperbolic Ricci-Bourguignon flow on a compact Riemannian manifold M coupled with the harmonic map flow between M and a fixed manifold N. At the first, we prove the unique short-time existence to solution of this system. Then, we find the second variational of some geometric structure of M along this system such as, curvature tensors. In addition, for emphasize the importance of Hyperbolic Ricci-Bourguignon flow, we give some examples of this flow on Riemannian manifolds.

Professor Themistocles M. Rassias’ special predilection and contribution to the study of mӧbius transformations is well known. möbius transformations of the open unit disc of the complex plane and, more generally, of the open unit ball of any real inner product space, give rise to möbius addition in the ball. The latter, in turn, gives rise to möbius gyro vector spaces that enable the Poincaré ball model of Hyperbolic geometry to be approached by gyro vector spaces, in full analogy with the common vector space approach to the standard model of Euclidean geometry. The purpose of this paper, dedicated to Professor Themistocles M. Rassias, is to employ the möbius gyro vector spaces for the introduction of the Hyperbolic square in the Poincare ball model of Hyperbolic geometry. We will find that the Hyperbolic square is richer in structure than its Euclidean counterpart.

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In this work we deals with almost automorphic behavior of solutions of a class of semilinear evolution equations. To achieve our goal we use interpolation theory and fixed point theory. As application, we examine sufficient conditions for existence of almost automorphic solutions of equations of the heat conduction theory.