Curvature & Connection Without Metric

In summary: It is independent of the connection used and can be calculated from the connection coefficients. However, for the Levi-Civita connection, the curvature tensor is particularly nice, being symmetric in the lower indices and antisymmetric in the upper indices. This is often referred to as "the" curvature tensor, but it is not unique. There are other curvature tensors that can be defined on a manifold with a connection.In summary, the two types of Christoffel symbols, ##\Gamma^a{}_{bc}## and ##\Gamma_{abc}##, have different meanings depending on the context. The first kind relates to the Levi-Civita connection on a Riemannian or pseudo-Riemannian manifold, while the second kind is
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In the absence of a metric, we can not raise and lower indices at will.

There are two sorts of Christoffel symbols, Christoffels of the first kind, ##\Gamma^a{}_{bc}## in component notation, and Christoffel symbols of the second kind, ##\Gamma_{abc}##. What's the relationship between the two kinds of Christoffel symbols? Is perhaps one of them a connection betwen vectors, and the other a connection between covectors?

Similarly, is ##R^a{}_{bcd}## "the" curvature tensor?

I suppose it'd be better to express this in terms of geometry rather than components, but I'm struggling a bit to do that.

This is all very basic, but I'm just not used to thinking about differential geometry without a metric :(.
 
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Typically the nomenclature ”Christoffel symbols” is generally reserved for the connection coefficients of the Levi-Civita connection. As such, they don't really hold meaning outside of a Riemannian or pseudo-Riemannian manifold.

The more general concept of connection coefficients ##\Gamma_{\mu i}^j## relate to the connection on a vector bundle where I have written the fiber indices with Latin letters. Choosing a basis ##E_i## for the fiber, the connection coefficients are defined by
$$
\nabla_\mu E_i = \Gamma_{\mu i}^j E_j.
$$
For a the tangent bundle, the indices are the same as the indices of the base manifold itself and you would write ##\Gamma_{\mu\nu}^\lambda## etc. (So if you don't want to think about general vector bundle, just replace ##i## and ##j## by Greek letters.)

The corresponding connection on the dual bundle is found by using a basis ##E^i## with the property ##E^i\cdot E_j = \delta^i_j##. From this follows that
$$
0 = \nabla_\mu (E^i \cdot E_j) = E^i \cdot \Gamma_{\mu j}^k E_k + (\nabla_\mu E^i)\cdot E_j = \Gamma_{\mu j}^i + (\nabla_\mu E^i)\cdot E_j
$$
leading to
$$
(\nabla_\mu E^i)\cdot E_j = -\Gamma_{\mu j}^i.
$$
In other words, the connection coefficients on the dual bundle is the same as those on the vector bundle with opposite sign.

As for the curvature tensor, it is uniquely defined by
$$
R(X,Y) A = \nabla_X \nabla_Y A - \nabla_Y \nabla_X A - \nabla_{[X,Y]} A,
$$
where ##X## and ##Y## are in the tangent space and ##A## in the fiber.
 
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FAQ: Curvature & Connection Without Metric

1. What is curvature?

Curvature refers to the degree to which a geometric shape deviates from being flat. In mathematics, it is measured by the amount of bending or twisting in a space or surface.

2. What is connection?

Connection is a mathematical concept that describes how points on a manifold (a space that locally resembles Euclidean space) are connected to each other. It is used to define the notion of parallel transport, which is important in understanding curvature.

3. How are curvature and connection related?

Curvature and connection are closely related because curvature is a measure of how a space or surface deviates from being flat, and connection is used to define the notion of parallel transport, which is essential in understanding curvature.

4. Can curvature and connection exist without a metric?

Yes, curvature and connection can exist without a metric. In fact, the concept of curvature and connection without a metric is important in understanding non-Euclidean geometries, such as those used in general relativity.

5. What are some real-world applications of curvature and connection without a metric?

Curvature and connection without a metric have applications in various fields, including physics, computer graphics, and differential geometry. They are used in understanding the behavior of spacetime in general relativity, creating realistic 3D models in computer graphics, and studying the properties of surfaces and manifolds in differential geometry.

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