Video 86 - Theorema Egregium

This paragraph introduces the topic of Theorema Egregium, a fundamental concept in tensor calculus that establishes a relationship between the Riemann curvature tensor and the extrinsic curvature tensor. The explanation begins with the application of the commutative operation to the shift tensor and expands on the meaning of this operation. The process involves using the rules from a previous video, identifying the need for a separate term for each surface index, and recognizing the negative term due to the position of the index omega. The paragraph also revisits an expression from video 74, which relates the covariant derivative of the shift tensor to the normal component and the curvature tensor, and demonstrates how to replace and rearrange terms to simplify the expression using the product rule.

The second paragraph continues the exploration of tensor calculus by focusing on finding the normal and surface projections of the equation derived in the previous paragraph. The normal projection involves contracting both sides of the equation with the normal vector, leading to the Kodama equation. This equation is significant as it is symmetric with respect to the first two indices, indicating that the expression is invariant under the swap of these indices. The paragraph also discusses the surface projection, which is achieved by contracting both sides with the inverse shift tensor. This process results in the Theorema Egregium, a remarkable theorem that connects extrinsic curvature information with intrinsic curvature, demonstrating that the Riemann tensor carries information about the manifold's curvature.

This paragraph delves into the significance of the Theorema Egregium, highlighting its importance in understanding the curvature of a manifold. The theorem shows that the Riemann tensor, an intrinsic object derived from the metric tensor, carries information about the curvature of the manifold. It contrasts this with the extrinsic curvature tensor, which is derived from the embedding of the manifold in the ambient space. The discussion then narrows down to two dimensions, examining a specific component of the Riemann tensor and relating it to the determinant of a 2x2 matrix, which is a scalar expression of the curvature tensor. The paragraph concludes by emphasizing that the Gaussian curvature, derived from the Riemann tensor, is an intrinsic property of the manifold.

The fourth paragraph further explains the intrinsic nature of Gaussian curvature, which can be computed directly from the metric tensor without any reference to extrinsic elements. It discusses how the Gaussian curvature is an invariant scalar expression and relates it to the determinant of the curvature tensor with mixed indices. The paragraph also introduces the concept of isometric manifolds, which have the same metric tensor but different embeddings, such as a plane and a cylinder. It uses the analogy of a contact lens to illustrate that bending a surface does not change its metric tensor or Gaussian curvature, emphasizing the theorem's remarkable property that curvature is an intrinsic measure.

This paragraph builds upon the previous discussions to derive additional relationships suitable for tensor calculus. It starts by simplifying the expression for the Riemann tensor in two dimensions and then generalizes it to eliminate hard-coded references, making it more adaptable for tensor calculus. The paragraph introduces the concept of Levi-Civita symbols and their relationship with the Riemann tensor, leading to an expression that relates the Gaussian curvature to the determinant of the curvature tensor with both indices in the upper and lower positions. It concludes by discussing the symmetries of the Riemann tensor and presenting a tensor expression that captures the intrinsic curvature information.

The final paragraph provides a recap of the video's main points, summarizing the key takeaways from the exploration of tensor calculus and the Theorema Egregium. It reiterates the significance of the Kodama equation and the Theorema Egregium, emphasizing the intrinsic nature of Gaussian curvature and its derivation from the Riemann tensor. The paragraph also reflects on the implications of the theorem, noting that while the Riemann tensor provides information about the curvature, it does not lead to a unique solution regarding the surface's embedding in the ambient space. It concludes with a quote from Gauss, highlighting the theorem's most remarkable aspect: the invariance of curvature under surface development.