Fundamental theorem of Riemannian geometry

In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric. Elliptic geometry is also sometimes called Riemannian geometry. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. In Differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a Moving frame around a curve In the mathematical field of Differential geometry, an affine connection is a geometrical object on a Smooth manifold which connects nearby Tangent In Riemannian geometry, the Levi-Civita connection is the torsion -free Riemannian connection, i Here a metric (or Riemannian) connection is a connection which preserves the metric tensor. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space

More precisely:

Let $(M, g)$ be a Riemannian manifold (or pseudo-Riemannian manifold) then there is a unique connection $\nabla$ which satisfies the following conditions:

for any vector fields $X, Y, Z$ we have $\partial_X(g(Y,Z))=g(\nabla_X Y,Z)+g(Y,\nabla_X Z)$ , where $\partial_X(g(Y,Z))$ denotes the derivative of the function $g (Y, Z)$ along vector field $X$ . In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold.

for any vector fields $X, Y$ we have $\nabla_XY-\nabla_YX=[X,Y]$ ,
where $[X, Y]$ denotes the Lie brackets for vector fields $X, Y$ . Lie bracket can refer to Lie algebra Lie bracket of vector fields In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space.

(The first condition expresses the fact that $\nabla g = 0$ , so that the metric tensor is preserved by parallel transport, while the second condition expresses the fact that the torsion $T^\nabla$ of $\nabla$ is zero. In Geometry, parallel transport is a way of transporting geometrical data along smooth curves in a Manifold. In Differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a Moving frame around a curve )

An extension of the fundamental theorem states that given a pseudo-Riemannian manifold there is a unique connection preserving the metric tensor with any given vector-valued 2-form as its torsion. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space

The following technical proof presents a formula for Christoffel symbols of the connection in a local coordinate system. In Mathematics, the covariant derivative is a way of specifying a Derivative along Tangent vectors of a Manifold. For a given metric this set of equations can become rather complicated. There are quicker and simpler methods to obtain the Christoffel symbols for a given metric, e. g. using the action integral and the associated Euler-Lagrange equations. In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation

Proof

In this proof we use Einstein notation. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational

Consider the local coordinate system $x^i,\ i=1,2,\dots,m=\dim(M)$ and let us denote by ${\mathbf e}_i={\partial\over\partial x^i}$ the field of basis frames.

The components $g_{i\;j}$ are real numbers of the metric tensor applied to a basis, i. e.

$g_{i j} \ \stackrel{\mathrm{def}}{=}\ {\mathbf g}({\mathbf e}_i,{\mathbf e}_j)$

To specify the connection it is enough to specify the Christoffel symbols $Γ k i j . In Mathematics, the covariant derivative is a way of specifying a Derivative along Tangent vectors of a Manifold.$

Since ${\mathbf e}_i$ are coordinate vector fields we have that

$[{\mathbf e}_i,{\mathbf e}_j]={\partial^2\over\partial x^j\partial x^i}-{\partial^2\over\partial x^i\partial x^j}=0$

for all $i$ and $j$ . Therefore the second property is equivalent to

$\nabla_{{\mathbf e}_i}{{\mathbf e}_j}-\nabla_{{\mathbf e}_j}{{\mathbf e}_i}=0,\ \$ which is equivalent to $\ \ \Gamma^k {}_{ij}=\Gamma^k {}_{ji}$ for all

i, j

and

k .

The first property of the Levi-Civita connection (above) then is equivalent to:

$\frac{\partial g_{ij}}{\partial x^k} = \Gamma^a {}_{k i} g_{aj} + \Gamma^a {}_{k j} g_{i a}.$

This gives the unique relation between the Christoffel symbols (defining the covariant derivative) and the metric tensor components.

We can invert this equation and express the Christoffel symbols with a little trick, by writing this equation three times with a handy choice of the indices

$\quad \frac{\partial g_{ij}}{\partial x^k} = +\Gamma^a {}_{ki} g_{aj} +\Gamma^a {}_{k j} g_{i a}$

$\quad \frac{\partial g_{ik}}{\partial x^j} = +\Gamma^a {}_{ji} g_{ak} +\Gamma^a {}_{jk} g_{i a}$

$- \frac{\partial g_{jk}}{\partial x^i} = -\Gamma^a {}_{ij} g_{ak} -\Gamma^a {}_{i k} g_{j a}$

By adding, most of the terms on the right hand side cancel and we are left with

$g_{i a} \Gamma^a {}_{kj} = \frac{1}{2} \left( \frac{\partial g_{ij}}{\partial x^k} +\frac{\partial g_{ik}}{\partial x^j} -\frac{\partial g_{jk}}{\partial x^i} \right)$

Or with the inverse of $\mathbf g$ , defined as (using the Kronecker delta)

$g^{k i} g_{i l}= \delta^k {}_l\,$

we write the Christoffel symbols as

$\Gamma^i {}_{kj} = \frac12 g^{i a} \left( \frac{\partial g_{aj}}{\partial x^k} +\frac{\partial g_{ak}}{\partial x^j} -\frac{\partial g_{jk}}{\partial x^a} \right)$

In other words, the Christoffel symbols (and hence the covariant derivative) are completely determined by the metric, through equations involving the derivative of the metric. In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two

The Koszul formula

An alternative proof of the Fundamental theorem of Riemannian geometry proceeds by showing that a torsion-free metric connection on a Riemannian manifold is necessarily given by the following formula, known as the Koszul formula:

$\begin{matrix} 2 g(\nabla_XY, Z) =& \partial_X (g(Y,Z)) + \partial_Y (g(X,Z)) - \partial_Z (g(X,Y))\\ {} & {}+ g([X,Y],Z) - g([X,Z],Y) - g([Y,Z],X). \end{matrix}$

This proves the uniqueness of the Levi-Civita connection. Existence is proven by showing that this expression is tensorial in X and Z, satisfies the Leibniz rule in Y, and that hence defines a connection. This is a metric connection, because the symmetric part of the formula in Y and Z is the first term on the first line; it is torsion-free because the anti-symmetric part of the formula in X and Y is the first term on the second line.

Proof

The Koszul formula

See also