We establish the weak continuity of the Gauss-Coddazi-Ricci system for isometric embedding with respect to the uniform L-bounded solution sequence for p > 2, which implies that the weak limit of the isometric embeddings of the manifold is still an isometric embedding. More generally, we establish a compensated compactness framework for the Gauss-Codazzi-Ricci system in differential geometry. Th...

where r = ∫ Rdμ/ ∫ dμ is the average scalar curvature (R is the scalar curvature) and Ric is the Ricci curvature tensor of h. Hamilton then spectacularly illustrated the success of this method by proving, when n = 3, that if the initial Riemannian metric has strictly positive Ricci curvature it evolves through time to a positively curved Einstein metric h∞ on M . And, because n = 3, such a Riem...

‎The object of the present paper is to introduce and study a type of non-flat semi-Riemannian manifolds‎, ‎called‎, ‎super generalized recurrent manifolds which generalizes both the notion of hyper generalized recurrent manifolds [‎A.A‎. ‎Shaikh and A‎. ‎Patra‎, On a generalized class of recurrent manifolds‎, Arch‎. ‎Math‎. ‎(Brno) 46 (2010) 71--78‎.] and weakly generalized recurrent manifolds ...

Let M^2n be a hoph hypersurfaces with parallel ricci operator and tangent to structure vector field in Sasakian space form. First, we show that structures and properties of hypersurfaces and hoph hypersurfaces in Sasakian space form. Then we study the structure of hypersurfaces and hoph hypersurfaces with a parallel ricci tensor structure and show that there are two cases. In the first case, th...

In the present study, we considered 3-dimensional generalized (κ, μ)-contact metric manifolds. We proved that a 3-dimensional generalized (κ, μ)-contact metric manifold is not locally φ-symmetric in the sense of Takahashi. However such a manifold is locally φ-symmetric provided that κ and μ are constants. Also it is shown that if a 3-dimensional generalized (κ, μ) -contact metric manifold is Ri...

For a complete Riemannian manifold M with (1,1)-elliptic Codazzi self-adjoint tensor field A, we use the divergence type operator $${L_A}(u): = div(A\nabla u)$$ and an extension of Ricci to extend some major comparison theorems in geometry. In fact such as mean curvature theorem, Bishop–Gromov volume Cheeger–Gromoll splitting theorem their famous topological consequences. Also get upper bound f...