Tensor Calculus Lecture 8: Embedded Surfaces and the Curvature Tensor

The script begins with an introduction to embedded surfaces and the curvature tensor, acknowledging that some topics have been skipped but will be backfilled later. The focus is on the concept of curvature, which is deemed essential and will be covered in several lectures. The speaker promises a brief introduction to the curvature tensor, which is eagerly anticipated by students. The prerequisites for understanding the lecture include a review of fundamental objects in Euclidean space, the Christoffel symbols, and the covariant derivative. The concept of an embedded surface in a three-dimensional Euclidean space is introduced, with the surface requiring two coordinates, s1 and s2, to be mapped.

The script continues with the concept of the position vector, which is an invariant field in the ambient space, and its restriction to the surface. The position vector R is now considered as a function of the surface coordinates s1 and s2. The covariant basis is defined by differentiating the position vector with respect to each variable, resulting in tangent vectors to the surface. The script also introduces the unit normal vector n, which is orthogonal to the covariant basis vectors and has a unit length. The importance of the normal vector in defining the tangential plane to the surface is highlighted.

The third paragraph delves into the formation of the covariant metric tensor through the dot product of the covariant basis elements. The metric tensor's role in measuring lengths, areas, and volumes on the surface is mentioned. The contravariant basis is introduced as the inverse of the covariant metric tensor, and the script provides an example using spherical coordinates to illustrate the covariant metric tensor's form. The importance of the metric tensor in various calculations on the surface is emphasized.

The script discusses the definition of Christoffel symbols, which are crucial for defining the covariant derivative. An attempt is made to define the Christoffel symbols using the covariant basis, but the presence of curvature introduces a problem, as the derivative of the covariant basis does not lie within the tangent plane. The script explains that this issue is a direct result of the surface's curvature and hints at the need for a new concept to address this problem.

The speaker presents two equivalent methods to define Christoffel symbols on surfaces, one intrinsic to the metric tensor and the other extrinsic, which uses the covariant basis. The intrinsic method relies on the metric tensor's derivatives, while the extrinsic approach maintains geometric intuition by working with the covariant basis. Both methods are shown to be valid for defining the Christoffel symbols and, by extension, the covariant derivative.

The script explores the covariant derivative's properties when applied to the covariant basis on a surface. It contrasts this with the Euclidean space, where the covariant derivative of the basis yields zeros. Due to curvature, the covariant derivative of the basis vectors on a surface is not zero and is instead proportional to the normal vector, with coefficients of proportionality denoted by D alpha beta. This observation leads to the introduction of the curvature tensor.

The final paragraph introduces the concept of mean curvature as a contraction of the curvature tensor, which has been identified as a natural consequence of the surface's curvature. The script emphasizes the tensor framework's ability to juggle indices and the symmetric properties of the curvature tensor. The mean curvature is presented as an important invariant property of the surface, derived from the curvature tensor.