Question

Let \(\mathbf{Z}=\xi^{k} \partial_{k}\). Show that $$ (\nabla \mathbf{Z})_{j}^{i} \equiv \xi_{; j}^{i}=\frac{\partial \xi^{i}}{\partial x^{j}}+\Gamma_{j k}^{i} \xi^{k} $$ and $$ \xi_{; l k}^{i}-\xi_{; k l}^{i}=R_{j k l}^{i} \xi^{j}+T_{l k}^{j} \xi_{; j}^{i} $$.

Short answer

After understanding the main concepts including the covariant derivative, and curvature and torsion tensors, and implementing these ideas, we first derived the expression for the covariant derivative as: \[ \xi_{; j}^{i}=\frac{\partial \xi^{i}}{\partial x^{j}}+\Gamma_{j k}^{i} \xi^{k} \] In the next step, we applied the definitions of the curvature and torsion tensors to get: \[ \xi_{; l k}^{i}-\xi_{; k l}^{i}=R_{j k l}^{i} \xi^{j}+T_{l k}^{j} \xi_{;j}^{i} \].

Step-by-step solution

Compute the Covariant Derivative

Starting with the first equation and using the definition of the covariant derivative, we compute the covariant derivative of the tensor field \( \xi^{i} \). The covariant derivative of a (1,0) tensor \(\xi^{i}\) is given by: \[\xi_{; j}^{i}=\frac{\partial \xi^{i}}{\partial x^{j}}+\Gamma_{j k}^{i} \xi^{k}\] Here \( \Gamma_{j k}^{i} \) is the Christoffel symbol, which are used to describe the geometric properties (like curvature) of the manifold, \( \xi^{k} \) are the components of the tensor and \(\partial_{x^{j}}\) denotes the partial derivative with respect to the coordinate \(x^{j}\).

Compute the Difference of Two Covariant Derivatives

In the second equation, apply the covariant derivative operation on the tensor \(\xi^{i}\) in two different orders to obtain the required expression: \[\xi_{; l k}^{i}-\xi_{; k l}^{i}\]

Apply the Definition of Curvature and Torsion Tensors

According to the definition of curvature and torsion tensors in Riemannian geometry, one finds: \[\xi_{; l k}^{i}-\xi_{; k l}^{i}=R_{j k l}^{i} \xi^{j}+T_{l k}^{j} \xi_{;j}^{i}\] Therein, \(R_{j k l}^{i}\) denotes the components of the Riemann-Christoffel tensor, a measure for the curvature of the underlying space, and \(T_{l k}^{j}\) represents the components of the torsion tensor, a measure for the inherent twist of the underlying space. These tensors are expressed in terms of the given covariant derivatives of the vector field \(\xi^{i}\).

Key concepts

Riemann-Christoffel tensor

The Riemann-Christoffel tensor, often simply called the Riemann tensor, is a fundamental concept in Riemannian geometry used to describe the curvature of a space. This mathematical object helps us understand how much a space deviates from being flat. In simpler terms, it measures how much the rules of geometry change as we move around in a space.

The tensor is denoted by the symbol \(R_{jkl}^{i}\), where the indices represent the particular components of the tensor. It is calculated based on the variations of the Christoffel symbols and is notably symmetrical, meaning that swapping positions of certain indices doesn't change its value. The Riemann-Christoffel tensor plays a critical role in the equations of General Relativity, which describe gravitational forces as curvature in space-time.

Understanding this tensor can be challenging due to its complexity, but a helpful way to think about it is through the lens of a curved surface like a sphere. When you walk along such a surface and return to your starting point, the direction you're facing might have changed, reflecting the curvature described by the Riemann-Christoffel tensor.

Christoffel symbol

Christoffel symbols, commonly represented by \(Gamma_{jk}^{i}\), are mathematical entities used in differential geometry, particularly in the study of Riemannian geometry. They are not tensors but play a crucial role as intermediary variables that express how vector components of a curve change when moving through a curved space.

These symbols help relate the derivatives of the metric tensor, which describes the distances in a given space, to the curvature. Essentially, they are a set of numbers that provide a way to compute covariant derivatives, thus allowing us to assess how vectors change without referring to a fixed coordinate system. This is fundamental when working with curved spaces, such as in the theory of General Relativity.

Although their expressions can be quite complex, involving second derivatives of the metric tensor, their use simplifies the description of how objects move in a curved space by making it possible to define parallel transport and geodesics.

torsion tensor

The torsion tensor is another key concept in differential geometry that complements the understanding of space's geometric properties. Unlike the Riemann-Christoffel tensor, which measures curvature, the torsion tensor gives insight into how a space twists.

Denoted by \(T_{lk}^{j}\), the torsion tensor is related to the Christoffel symbols, which typically have symmetries that make them zero in spaces without torsion. This tensor reflects the twist or asymmetry in the connection of a space. In simpler terms, while curvature tells us how much a space bends, torsion tells us how it swivels.

Torsion is an essential concept in some advanced gravitational theories, beyond the scope of standard Riemannian geometry. In these theories, torsion can represent intrinsic spin of matter or affect the parallel transport of vectors. Understanding torsion involves recognizing that not all geometric transformations are related simply to curvature, as twisting is another dimension.

Riemannian geometry

Riemannian geometry is a branch of differential geometry focused on the study of curved surfaces or spaces. Named after the mathematician Bernhard Riemann, this field generalizes the concepts of geometry to allow for curved, rather than strictly flat, surfaces.

At its core, Riemannian geometry uses a Riemannian metric, a type of tensor that defines the length of vectors and angles between them in a curved space. This metric gives rise to the concepts of distance and curvature, allowing us to apply calculus to spaces of arbitrary curvature.

Key features of Riemannian geometry include the use of Christoffel symbols and the Riemann-Christoffel tensor to describe curvature, helping us understand gravitational phenomena in the language of geometry. This framework is central to Einstein's theory of General Relativity, where space-time is molded by mass and energy. For students, grasping these abstract concepts can be aided by visual aids and understanding the analogies of simple curved objects like spheres or hyperbolic surfaces.