Levi-Civita connection

In mathematics, especially in the Riemannian geometry, a branch of differential geometry, is meant by a Levi- Civita connection is a connection on the tangent bundle of a Riemannian or semi- Riemannian manifold, which is compatible with the metric of the manifold in a certain way. The Levi- Civita connection plays a central role in the modern design of the Riemannian geometry. It is there is a generalization of the conventional directional derivative of the multi-dimensional Euclidean space in differential calculus, and is suitable to quantify the change in direction of a vector field in the direction of a further vector field. The term of the Levi -Civita connection is equivalent to the parallel transport in the sense of Levi -Civita and therefore a means to set the tangent at different points related to each other, where the name comes context. Since the (semi-) Riemannian geometry is an essential tool for the formulation of general relativity, the Levi- Civita connection is used here. A further application is the Levi- Civita connection with the construction of the Dirac operator a spinning manifold.

Motivation

For vector fields and on the Euclidean space we define the Levi- Civita connection as the directional derivative of Y to X, ie the directional derivative of the individual components from Y to X:

The usual directional derivative referred.

If a submanifold, and the vector fields are then defined to be a vector field, but whose images are not necessary in the tangent space of, in the tangent space of. For each but you can use the orthogonal projection defined and then

This relation satisfies the axioms given below, after the main theorem of differential geometry he agrees with the Levi- Civita connection coincide. The advantage of the below axiomatic access is that you can consider the Levi- Civita connection of a Riemannian manifold independent of a to be chosen embedding.

Definition

There is a (semi-) Riemannian manifold. Then there exists a unique connection on the tangent bundle of the following characteristics:

• Is torsion, that is, it is for all vector fields. It denotes the Lie bracket of vector fields and.
• Is a metric connection, that is, it is for all vector fields, and.

This connection is called the Levi- Civita connection of the Riemannian connection or even of. It is named after Tullio Levi -Civita.

Properties

Fundamental theorem of Riemannian geometry

From the above definition is not clear whether such a Levi- Civita connection exists. This has to be proved so. The claim that such a connection exists and is also clearly, is frequently mentioned in the literature Fundamental Theorem of Riemannian geometry. The Levi- Civita connection is in fact an essential tool for the construction of the Riemannian curvature theory. Because of the curvature tensor is defined by means of a link, so it makes sense to use the clearly excellent Levi- Civita connection for the definition of the Riemann curvature tensor in Riemannian geometry.

Koszul formula

The Levi- Civita connection is uniquely described by the Koszul formula (named after Jean -Louis Koszul )

This is an implicit, global description of which is suitable especially for an abstract existence proof of. It can be from a local description of the construction of well.

Christoffel symbols

A local description of obtained as follows. Generally, a correlation is described locally on a vector bundle with its connection coefficients. The connection coefficients of the Levi -Civita connection are the classical Christoffel symbols of the second kind. This means in particular that of relating a map

With

Applies. This is the inverse matrix of the Riemannian fundamental tensor and the coordinate basis of the map.

Since the Levi- Civita connection is torsion-free, the Christoffel symbols are symmetric, ie for all, and is: .

This is called the covariant derivative of along, since the classical covariant derivative of the tensor calculus of Gregorio Ricci and Tullio Levi -Civita Curbastro generalized.

Relations with the directional derivative

Let a (semi-) Riemannian manifold and the Levi- Civita connection of. Also, be, vector fields on. Then can be summarized as follows as a generalization of the concept of directional derivative for vector fields of understand.

• It is a point. Then depends only on the tangent vector and the vector field. If one chooses a smooth curve with and and is denoted by the parallel transport along in the sense of Levi -Civita, it shall

• It is a point. Then a map exists around so that the metric fundamental tensor with respect to the point given by (normal coordinates). With respect to such card is valid at the point

The Levi- Civita connection has a particularly simple description in the case in which is a Riemannian manifold which arises from that one. Standard metric of a submanifold of limits In this case, the Levi - Civita connection of is given as follows. It is

Here are vector fields on, continuations of these vector fields to vector fields on completely, the directional derivative of along the vector field and the orthogonal projection of the tangent space with base point.

Directional derivative along curves

The Levi- Civita connection allows the concept of acceleration of a smooth curve that define extends in a Riemannian manifold. This leads to a description of the geodesics of the underlying Riemannian manifold as the acceleration- free curves. First, the Levi- Civita connection defined ( just as every connection on a vector bundle) a directional derivative for vector fields are explained along a curve. This directional derivative measure the rate of change of the vector field in the direction of the curve. There are different names for this derivative in use. We call the most common after the definition.

It should be a smooth curve in the Riemannian manifold and a vector field along. The directional derivative of along the point is

Other common names for this size are

In particular, the velocity field of even a vector field along the curve. The acceleration of is along the vector field. The curve is then exactly one geodesic Riemannian manifold, when their acceleration disappears. From a physical point of view, ie geodesics can be kinematically than the curves indicate where a particle would follow in the Riemannian manifold, if it is not subjected to force.

Parallel transport

In general, a parallel transport along a curve defined with respect to a link on a beam vector an isomorphism between the fibers which are base points on the curve. Is the connection of the Levi- Civita connection of a Riemannian manifold, then the isomorphisms are orthogonal, ie, length and angle preserving. The current induced by the Levi- Civita connection of a Riemannian manifold parallel transport agrees with the 1918 first defined by Levi -Civita parallel transport agreement ( cf. parallel transport in the sense of Levi -Civita ). This was anticipated in a special case of Ferdinand Minding.

Riemannian connection

In the theory of principal bundles relationships are defined as Lie algebra -valued 1-forms. Since the frame bundle of a Riemannian manifold is a principal bundle with the group, you can define a connection form with the help of the Levi -Civita connexion as follows.

Are coordinates in a local neighborhood, so that the base is an element of the frame beam, ie. The Christoffel symbols of the Levi -Civita connection are then described by. The by defined -valued 1-form on has the decomposition in these coordinates. Be the continued on a base of neighborhoods. Then defined

A matrix -valued 1-form and it is

The point defined by the Riemannian connection, parallel transport on the frame bundle is consistent with the defined by the Levi- Civita connection on the tangent bundle of parallel transport.