Decompositions of curvature tensor

It is known that Riemann curvature tensor and the related curvatures such as Ricci curvature, scalar curvature and sectional curvature are fundamental tools while studying the geometry and topology of a manifold. Firstly we introduce these curvatures in this study. The space of tensors of type (0,4) whose elements have same algebraic symmetry properties of the Riemann curvature tensor is called Space of Curvature Like Tensors. In this thesis the decompositions of the Space of Curvature Like Tensors on a Riemann manifold is investigated. According to the decomposition of Space of Curvature Like Tensors, the Riemann curvature tensor is decomposed as Rm=Um+Zm+Wm. The part Wm in this decomposition is called Weyl Tensor. Some equalities and Theorems related to the parts of the decomposition of Riemann curvature tensor are given. Lastly, it is shown that the Weyl Tensor is conformally invariant.