Video 81 - Intrinsic Curvature

This paragraph introduces the concept of re-examining surface curvature from an intrinsic perspective, contrasting it with the extrinsic approach previously discussed. The intrinsic approach focuses on the properties of the surface itself, without reliance on the surrounding space. The video aims to show that Gaussian curvature can be derived directly from the surface's intrinsic properties, rather than using the extrinsic curvature tensor. The extrinsic curvature tensor is defined using external objects like the tangent plane and normal vector, which are related to the surface's embedding in the ambient space. The paragraph also introduces the idea that Gaussian curvature is an intrinsic property, suggesting that it can be understood without reference to any extrinsic objects, and sets the stage for an intrinsic analysis of surface curvature in subsequent videos.

This paragraph uses the analogy of two-dimensional creatures, Bert and Ernie, living on a spherical surface to illustrate the intrinsic perspective of curvature. The creatures can measure distances, angles, and areas on the surface but are unaware of the third dimension or the ambient space. The video script describes how these creatures would measure a circle's circumference and area, expecting results based on the formulas for a flat plane. However, as the circle grows larger, discrepancies between the expected and actual measurements emerge, indicating the presence of curvature. The script explains that the deviation from the expected Euclidean results is a clue to the intrinsic curvature of the surface, with positive curvature leading to calculated values larger than actual measurements and vice versa for negative curvature.

The script explores the properties of triangles on curved surfaces, specifically focusing on the sum of angles and the law of cosines. Initially, the triangle appears to conform to Euclidean geometry, with angles summing to 180 degrees and the law of cosines accurately predicting side lengths. However, as the triangle is enlarged, deviations occur: the angles sum to more than 180 degrees, and the law of cosines no longer matches the actual measurements. This section emphasizes that in a curved manifold, the local properties can appear Euclidean, but as the scale increases, the intrinsic curvature becomes evident, leading to deviations from flat space geometry. The use of geodesics, the shortest paths between points on the surface, is highlighted to explain why the angles in a large triangle on a sphere add up to more than 180 degrees.

This paragraph discusses the concept of parallel lines in the context of curved spaces, using the example of Bert and Ernie walking on a curved surface. In Euclidean geometry, parallel lines never intersect, but on a curved surface, such as a sphere, this is not the case. The script shows that even when the characters attempt to walk in parallel paths, their paths eventually intersect due to the curvature of the surface. This leads to a discussion of Euclid's fifth postulate, which states that through a point not on a given line, there is exactly one line parallel to the given line. The script explains that this postulate only holds true in flat spaces and that in curved spaces, the concept of parallel lines must be redefined. The video concludes by emphasizing the differences between flat and curved spaces and the implications for geometric proofs and theorems.