# Covariant derivatives, connections, metrics, and Christoffel symbols

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TL;DR Summary
How are these concepts related?
Is a connection the same thing as a covariant derivative in differential geometry?

What Is the difference between a covariant derivative and a regular derivative?

If you wanted to explain these concepts to a layperson, what would you tell them?

If you wanted to explain these concepts to a layperson, what would you tell them?

Summary:: How are these concepts related?

Is a connection the same thing as a covariant derivative in differential geometry?
A connection on a manifold induces a connection on its tangent bundle which is called covariant derivative. Hence the answer is no, since they are defined differently, and the answer is yes, since the covariant derivative of the tangent bundle restricted to the manifold is the original connection.
What Is the difference between a covariant derivative and a regular derivative?
A covariant derivative applies to tensor products, a regular derivative does not.
If you wanted to explain these concepts to a layperson, what would you tell them?
Any derivative is a directional derivative, if you only increase the degree of abstraction. The concept of a directional derivative is successively generalized to wider ranges of applications.

A directional derivative at a certain point of a curved space is a tangent. All tangents at this point define the tangent space. All tangent spaces (over all possible points) define a tangent bundle. A connection is a tool to make tangents of a curved space at different points comparable. Different connections lead to different comparisons.

You cannot explain everything to a layperson in detail, because some technical definitions are necessary. If you don't mind using them, you can find an overview here:
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/

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A covariant derivative can be thought of as a generalization of the idea of a directional derivative of a vector field in multivariable calculus.

If one has a covariant derivative at every point of a manifold and if these vary smoothly then one has an affine connection. Smoothly means that for two smooth vector fields the covariant derivatives of one with respect to the other at each point also form a smooth vector field.

So an affine connection is a smooth choice of covariant derivatives at the points of the manifold.

Example: Suppose one has a manifold that is embedded in Euclidean space. A vector field tangent to the manifold is also tangent to Euclidean space so one can take its usual directional derivative . In general this derivative will have a normal component and will not be a vector tangent the manifold. So it is not a covariant derivative on the tangent bundle of the manifold. But the orthogonal projection of this directional derivative onto the tangent space of the manifold is a covariant derivative. It is the part of the directional derivative that an observer living on the manifold can see without looking up outside of his world. A different embedding will yield a different projection i.e. a different covariant derivative since the shape of the manifold will be different. One can show that every Levi-Civita connection on a closed Riemannian manifold can be obtained in this way. This picture illustrates why Levi-Civita connections are generalizations of the idea of directional derivative.

For a layman one might give the specific example of a great circle on the sphere viewed as a unit speed path i.e. as a path of uniform circular motion. The directional derivative of its unit speed velocity vector with respect to its velocity vector at a point is just its acceleration and this is perpendicular to the circle and to the sphere. So the orthogonal projection onto the tangent plane of the sphere i.e. its covariant derivative is zero. To an antman living on the sphere, this path will have zero acceleration and will look like a straight line at least for short distances. But what a surprise when he returns to where he started.

More generally if one can slice a surface in 3 space with a plane that is everywhere perpendicular to the surface then the curve of intersection will appear like a straight line to an antman. Try doing this with a cylinder or a symmetrical torus.

In a general setting the vector field ##Y## does not have to be tangent to the manifold. It can reside in a smooth vector bundle. Affine connections and covariant derivatives make sense for smooth vector bundles over smooth manifolds. Given a tangent vector ##X_{p}## at the point ##p## and a vector field ##Y## in some vector bundle ##E## then the covariant derivative of ##Y## with respect to ##X_{p}## is a new vector in ##E## in the fiber above ##p##.

The term "connection" has other meanings also and these may not be defined in terms of covariant derivatives although the ideas are related. You might look at connections on principal Lie group bundles for starters.

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You might supplement the example of a great circle on a sphere with a longitudinal circle. In this case the directional derivative of the unit speed vector is not perpendicular to the sphere and has a non-zero tangential component. The antman moving at constant speed along this circle will feel an acceleration perpendicular to his circular path of motion and pointing to one of the poles.