Tensor Calculus Lecture 11a: Gauss' Theorema Egregium, Part 1

The speaker expresses enthusiasm for discussing Gus's theorem, a topic they find particularly exciting. Despite being eager to end the lecture due to a sore throat, they are determined to cover the theorem, which they believe will be a highlight of the series. The significance of the theorem is mentioned as extending beyond the lecture's scope, touching on areas like topology and relativity. The lecture aims to derive the theorem, starting with a correction of a previous calculation mistake and moving on to warm-up exercises before delving into the main derivation. The setting for the calculations will be a general one, considering surfaces embedded in a non-Euclidean space.

The paragraph focuses on the mathematical derivation of the relationship between the surface Laplacian and the ambient Laplacian. The process involves applying the chain rule and considering the normal and covariant derivatives. The speaker discusses the importance of mean curvature in the derivation and how it emerges as a significant factor. The final expression obtained relates the surface Laplacian to the ambient Laplacian, with additional terms involving the normal derivative and second-order normal derivative of a function U on the surface.

This section delves into the relationship between surface and ambient Christoffel symbols. The speaker uses the shift tensor to relate the basis elements of the surface to those of the ambient space. The derivation involves applying the product rule and considering the chain rule for differentiation. The final result shows how the surface Christoffel symbols are related to the ambient ones, with a projection term and an additional term that vanishes in Euclidean spaces but is significant in non-Euclidean contexts.

The speaker introduces the concept of the commutator for tensors with ambient indices, such as the normal or shift tensor. They discuss the implications of the ambient space not being Euclidean and how this affects the calculation of the Christoffel symbols. The paragraph sets the stage for a deeper exploration of the commutator in the context of non-Euclidean spaces and its role in the derivation of Gauss's theorem.

The paragraph explores the implications of working in a non-Euclidean ambient space, where the metric tensor is arbitrarily imposed and does not necessarily satisfy the conditions that would make the Riemann Christoffel symbol vanish. The speaker invites the audience to consider spaces where the metric tensor does not arise from a dot product, leading to a different kind of space known as a Riemannian space. This discussion is crucial for understanding the broader context of the derivation of Gauss's theorem.

The speaker presents a shortcut to derive the commutator for tensors defined in the entire ambient space, even though the derivation is not fully valid for objects defined only on the surface. The calculation involves using the chain rule and considering the product rule. The final result is a boxed expression that relates the commutator to the Riemann Christoffel tensor, providing a crucial step towards deriving Gauss's theorem.

The final paragraph prepares the audience for the main event: the derivation of Gauss's theorems. The speaker acknowledges that the previous derivation, while not entirely valid, leads to the correct final answer. They encourage the audience to consider the implications for covariant indices and combinations of surface and ambient indices, hinting at the complexity of the upcoming derivation. The speaker then pauses to erase the board, signaling a transition to the next phase of the lecture.