How do we Describe and Understand Spacetime Curvature?

Spacetime Curvature is a fundamental concept in Cosmology and Relativity. There is no way to describe space and time, and the evolution of the Universe since the Big Bang without it. In this introduction to Curvature, I detail the mathematical framework that underpins a space that is homgeneous and isotropic, with a globally uniform constant curvature. I discuss in detail various metrics, and how we measure lengths and areas in such spaces. I also discuss the real machinery of spacetime: tensorial measures of the space using the Riemann Curvature Tensor and its related components.
Most importantly, I try, hopefully not desperately, to demonstrate that there is no reason or need for spacetime to curve "into" anything. Spacetime curvature is a property of the space itself and doesn't a priori demand or dictate extra physical dimensions to accomodate the curvature.
This is the ninth of the videos in my new series of Cosmology. I'm going through Dr. Barbara Ryden's textbook "Introduction to Cosmology". If you follow along, you'll get a full upper-division undergraduate course in Cosmology. I used this textbook at William Paterson University.
This course will cover the current state of the science of Cosmology. To follow along, it'll be a good idea for you to ge to know your calculus. Here are the topics of this video:
Introductory Cosmology
Chapter 03: Newton versus Einstein
Section 04: Describing Curvature
Some things covered:
Textbook: / introduction-to-cosmology
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