Abstract
In this chapter we introduce the important concept of the fourth-order curvature tensor in the Riemannian metric spaces. The covariant and mixed fourth-order Curvature tensors are the main ingredients of the study of differential geometry, general theory of relativity and cosmology. We first define the curvature tensor and study its properties in great detail, as these properties are extensively used in the rest of the book. Thereafter, we define two more essential mathematical objects, that is, the second-order Ricci tensor and the zeroth-order Ricci scalar, which are both derived from the mixed fourth-order curvature tensor by contractions with the metric tensor. Finally, as a simple example, we study the curvature properties of a unit sphere.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
