Γεωμετρία Riemann

Γεωμετρία Riemann

Riemannian Geometry

Εφαπτόμενος Χώρος Συνεφαπτόμενος Χώρος

- Ένας Επιστημονικός Κλάδος της Γεωμετρίας.

Ετυμολογία

Το όνομα "Riemannian" σχετίζεται ετυμολογικά με το όνομα "Riemann".

Christoffel symbols, covariant derivative

In a smooth coordinate chart, the Christoffel symbols are given by:

$\Gamma^m_{ij}=\frac12 g^{km} \left( \frac{\partial}{\partial x^i} g_{kj} +\frac{\partial}{\partial x^j} g_{ik} -\frac{\partial}{\partial x^k} g_{ij} \right)$

Here $g^{ij}$ is the inverse matrix to the metric tensor $g_{ij}$. In other words,

$\delta^i_j = g^{ik}g_{kj}$

and thus

$n = \delta^i_i = g^i_i = g^{ij}g_{ij}$

is the dimension of the manifold.

Christoffel symbols satisfy the symmetry relation

$\Gamma^i_{jk}=\Gamma^i_{kj}$

which is equivalent to the torsion-freeness of the Levi-Civita connection.

The contracting relations on the Christoffel symbols are given by

$\Gamma^i_{ki}=\frac{1}{2} g^{im}\frac{\partial g_{im}}{\partial x^k}=\frac{1}{2g} \frac{\partial g}{\partial x^k} = \frac{\partial \log \sqrt{|g|}}{\partial x^k} \$

and

$g^{k\ell}\Gamma^i_{k\ell}=\frac{-1}{\sqrt{|g|}} \;\frac{\partial\left(\sqrt{|g|}\,g^{ik}\right)} {\partial x^k}$

where |g| is the absolute value of the determinant of the metric tensor $g_{ik}\$. These are useful when dealing with divergences and Laplacians (see below).

The covariant derivative of a vector field with components $v^i$ is given by:

$v^i {}_{;j}=\nabla_j v^i=\frac{\partial v^i}{\partial x^j}+\Gamma^i_{jk}v^k$

and similarly the covariant derivative of a $(0, 1)$-tensor field with components $v_i$ is given by:

$v_{i;j}=\nabla_j v_i=\frac{\partial v_i}{\partial x^j}-\Gamma^k_{ij} v_k$

For a $(2,0)$-tensor field with components $v^{ij}$ this becomes

$v^{ij}_{;k}=\nabla_k v^{ij}=\frac{\partial v^{ij}}{\partial x^k} +\Gamma^i_{k\ell}v^{\ell j}+\Gamma^j_{k\ell}v^{i\ell}$

and likewise for tensors with more indices.

The covariant derivative of a function (scalar) $\phi$ is just its usual differential:

$\nabla_i \phi=\phi_{;i}=\phi_{,i}=\frac{\partial \phi}{\partial x^i}$

Because the Levi-Civita connection is metric-compatible, the covariant derivatives of metrics vanish,

$\nabla_k g_{ij} = \nabla_k g^{ij} = 0$

The geodesic $X(t)$ starting at the origin with initial speed $v^i$ has Taylor expansion in the chart:

$X(t)^i=tv^i-\frac{t^2}{2}\Gamma^i_{jk}v^jv^k+O(t^2)$

Curvature tensors

Riemann curvature tensor

If one defines the curvature operator as $R(U,V)W=\nabla_U \nabla_V W - \nabla_V \nabla_U W -\nabla_{[U,V]}W$ and the coordinate components of the $(1,3)$-Riemann curvature tensor by $(R(U,V)W)^\ell={R^\ell}_{ijk}W^iU^jV^k$, then these components are given by:

${R^\ell}_{ijk}= \frac{\partial}{\partial x^j} \Gamma_{ik}^\ell-\frac{\partial}{\partial x^k}\Gamma_{ij}^\ell +\sum^{n}_{s=1}(\Gamma_{js}^\ell\Gamma_{ik}^s-\Gamma_{ks}^\ell\Gamma_{ij}^s)$

where n denotes the dimension of the manifold. Lowering indices with $R_{\ell ijk}=g_{\ell s}{R^s}_{ijk}$ one gets

$R_{ik\ell m}=\frac{1}{2}\left( \frac{\partial^2g_{im}}{\partial x^k \partial x^\ell} + \frac{\partial^2g_{k\ell}}{\partial x^i \partial x^m} - \frac{\partial^2g_{i\ell}}{\partial x^k \partial x^m} - \frac{\partial^2g_{km}}{\partial x^i \partial x^\ell} \right) +g_{np} \left( \Gamma^n_{k\ell} \Gamma^p_{im} - \Gamma^n_{km} \Gamma^p_{i\ell} \right). \$

The symmetries of the tensor are

$R_{ik\ell m}=R_{\ell mik}\$ and $R_{ik\ell m}=-R_{ki\ell m}=-R_{ikm\ell}.\$

That is, it is symmetric in the exchange of the first and last pair of indices, and antisymmetric in the flipping of a pair.

The cyclic permutation sum (sometimes called first Bianchi identity) is

$R_{ik\ell m}+R_{imk\ell}+R_{i\ell mk}=0.\$

The (second) Bianchi identity is

$\nabla_m R^n {}_{ik\ell} + \nabla_\ell R^n {}_{imk} + \nabla_k R^n {}_{i\ell m}=0,\$

that is,

$R^n {}_{ik\ell;m} + R^n {}_{imk;\ell} + R^n {}_{i\ell m;k}=0 \$

which amounts to a cyclic permutation sum of the last three indices, leaving the first two fixed.

Ricci and scalar curvatures

Ricci and scalar curvatures are contractions of the Riemann tensor. They simplify the Riemann tensor, but contain less information.

The Ricci curvature tensor is essentially the unique nontrivial way of contracting the Riemann tensor:

$R_{ij}={R^\ell}_{i\ell j}=g^{\ell m}R_{i\ell jm}=g^{\ell m}R_{\ell imj} =\frac{\partial\Gamma^\ell_{ij}}{\partial x^\ell} - \frac{\partial\Gamma^\ell_{i\ell}}{\partial x^j} + \Gamma^\ell_{ij} \Gamma^m_{\ell m} - \Gamma^m_{i\ell}\Gamma^\ell_{jm}.\$

The Ricci tensor $R_{ij}$ is symmetric.

By the contracting relations on the Christoffel symbols, we have

$R_{ik}=\frac{\partial\Gamma^\ell_{ik}}{\partial x^\ell} - \Gamma^m_{i\ell}\Gamma^\ell_{km} - \nabla_k\left(\frac{\partial}{\partial x^i}\left(\log\sqrt{|g|}\right)\right).\$

The scalar curvature is the trace of the Ricci curvature,

$R=g^{ij}R_{ij}=g^{ij}g^{\ell m}R_{i\ell jm}$.

The "gradient" of the scalar curvature follows from the Bianchi identity (proof):

$\nabla_\ell R^\ell {}_m = {1 \over 2} \nabla_m R, \$

that is,

$R^\ell {}_{m;\ell} = {1 \over 2} R_{;m}. \$

Einstein tensor

The Einstein tensor G^ab is defined in terms of the Ricci tensor R^ab and the Ricci scalar R,

$G^{ab} = R^{ab} - {1 \over 2} g^{ab} R \$

where g is the metric tensor.

The Einstein tensor is symmetric, with a vanishing divergence (proof) which is due to the Bianchi identity:

$\nabla_a G^{ab} = G^{ab} {}_{;a} = 0. \$

Weyl tensor

The Weyl tensor is given by

$C_{ik\ell m}=R_{ik\ell m} + \frac{1}{n-2}\left( - R_{i\ell}g_{km} + R_{im}g_{k\ell} + R_{k\ell}g_{im} - R_{km}g_{i\ell} \right) + \frac{1}{(n-1)(n-2)} R \left( g_{i\ell}g_{km} - g_{im}g_{k\ell} \right),\$

where $n$ denotes the dimension of the Riemannian manifold.

Gradient, divergence, Laplace–Beltrami operator

The gradient of a function $\phi$ is obtained by raising the index of the differential $\partial_i\phi dx^i$, that is:

$\nabla^i \phi=\phi^{;i}=g^{ik}\phi_{;k}=g^{ik}\phi_{,k}=g^{ik}\partial_k \phi=g^{ik}\frac{\partial \phi}{\partial x^k}$

The divergence of a vector field with components $V^m$ is

$\nabla_m V^m = \frac{\partial V^m}{\partial x^m} + V^k \frac{\partial \log \sqrt{|g|}}{\partial x^k} = \frac{1}{\sqrt{|g|}} \frac{\partial (V^m\sqrt{|g|})}{\partial x^m}.\$

The Laplace–Beltrami operator acting on a function $f$ is given by the divergence of the gradient:

$\begin{align} \Delta f &= \nabla_i \nabla^i f = \frac{1}{\sqrt{\det g}} \frac{\partial }{\partial x^j}\left(g^{jk}\sqrt{\det g}\frac{\partial f}{\partial x^k}\right) \\ &= g^{jk}\frac{\partial^2 f}{\partial x^j \partial x^k} + \frac{\partial g^{jk}}{\partial x^j} \frac{\partial f}{\partial x^k} + \frac12 g^{jk}g^{il}\frac{\partial g_{il}}{\partial x^j}\frac{\partial f}{\partial x^k} = g^{jk}\frac{\partial^2 f}{\partial x^j \partial x^k} - g^{jk}\Gamma_{jk}^l\frac{\partial f}{\partial x^l} \end{align}$

The divergence of an antisymmetric tensor field of type $(2,0)$ simplifies to

$\nabla_k A^{ik}= \frac{1}{\sqrt{|g|}} \frac{\partial (A^{ik}\sqrt{|g|})}{\partial x^k}.\$

Kulkarni–Nomizu product

The Kulkarni–Nomizu product is an important tool for constructing new tensors from existing tensors on a Riemannian manifold. Let $h$ and $k$ be symmetric covariant 2-tensors. In coordinates,

$h_{ij} = h_{ji} \qquad \qquad k_{ij} = k_{ji}$

Then we can multiply these in a sense to get a new covariant 4-tensor, which we denote $h \odot k$. The defining formula is

$\left(h\odot k\right)_{ijkl} = h_{ik}k_{jl} + h_{jl}k_{ik} - h_{il}k_{jk} - h_{jk}k_{il}$

Often the Kulkarni–Nomizu product is denoted by a circle with a wedge that points up inside it. However, we will use $\odot$ instead throughout this article. Clearly, the product satisfies

$h \odot k = k \odot h$

Let us use the Kulkarni–Nomizu product to define some curvature quantities.

Weyl tensor

The Weyl tensor $W_{ijkl}$ is defined by the formula

$R_{ijkl} = -\frac{R}{2n(n-1)}(g\odot g)_{ijkl} + \frac{1}{n-2}\left[ \left(Ric -\frac{R}{n}g\right) \odot g \right]_{ijkl} + W_{ijkl}$

Each of the summands on the righthand side have remarkable properties. Recall the first (algebraic) Bianchi identity that a tensor $T_{ijkl}$ can satisfy:

$T_{ijkl} + T_{kijl} + T_{jkil} = 0$

Not only the Riemann curvature tensor on the left, but also the three summands on the right satisfy this Bianchi identity. Furthermore, the first factor in the second summand has trace zero. The Weyl tensor is a symmetric product of alternating 2-forms,

$W_{ijkl} = -W_{jikl} \qquad W_{ijkl} = W_{klij}$

just like the Riemann tensor. Moreover, taking the trace over any two indices gives zero,

$W^i_{jki} = 0$

In fact, any tensor that satisfies the first Bianchi identity can be written as a sum of three terms. The first, a scalar multiple of $g \odot g$. The second, as $H \odot g$ where $H$ is a symmetric trace-free 2-tensor. The third, a symmetric product of alternating two-forms which is totally traceless, like the Weyl tensor described above.

The most remarkable property of the Weyl tensor, though, is that it vanishes ($W = 0$)if and only if a manifold $M$ of dimension $n \geq 4$ is locally conformally flat. In other words, $M$ can be covered by coordinate systems in which the metric $ds^2$ satisfies

$ds^2 = f^2\left(dx_1^2 + dx_2^2 + \ldots dx_n^2\right)$

This is essentially because $W^i_{jkl}$ is invariant under conformal changes.

In an inertial frame

An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations $g_{ij} = \delta_{ij}$ and $\Gamma^i_{jk}=0$ (but these may not hold at other points in the frame). In such a frame, the expression for several operators is simpler. Note that the formulae given below are valid at the origin of the frame only.

$R_{ik\ell m}=\frac{1}{2}\left( \frac{\partial^2g_{im}}{\partial x^k \partial x^\ell} + \frac{\partial^2g_{k\ell}}{\partial x^i \partial x^m} - \frac{\partial^2g_{i\ell}}{\partial x^k \partial x^m} - \frac{\partial^2g_{km}}{\partial x^i \partial x^\ell} \right)$

Under a conformal change

Let $g$ be a Riemannian metric on a smooth manifold $M$, and $\varphi$ a smooth real-valued function on $M$. Then

$\tilde g = e^{2\varphi}g$

is also a Riemannian metric on $M$. We say that $\tilde g$ is conformal to $g$. Evidently, conformality of metrics is an equivalence relation. Here are some formulas for conformal changes in tensors associated with the metric. (Quantities marked with a tilde will be associated with $\tilde g$, while those unmarked with such will be associated with $g$.)

$\tilde g_{ij} = e^{2\varphi}g_{ij}$

$\tilde \Gamma^k_{ij} = \Gamma^k_{ij}+ \delta^k_i\partial_j\varphi + \delta^k_j\partial_i\varphi-g_{ij}\nabla^k\varphi$

Note that the difference between the Christoffel symbols of two different metrics always form the components of a tensor.

$d\tilde V = e^{n\varphi}dV$

Here $dV$ is the Riemannian volume element.

$\tilde R_{ijkl} = e^{2\varphi}\left( R_{ijkl} - \left[ g \odot \left( \nabla\partial\varphi - \partial\varphi\partial\varphi + \frac{1}{2}\|\nabla\varphi\|^2g \right)\right]_{ijkl} \right)$

Here $\odot$ is the Kulkarni-Nomizu product defined earlier in this article. The symbol $\partial_k$ denotes partial derivative, while $\nabla_k$ denotes covariant derivative.

$\tilde R_{ij} = R_{ij} - (n-2)\left[ \nabla_i\partial_j \varphi - (\partial_i \varphi)(\partial_j \varphi) \right] + \left( \triangle \varphi - (n-2)\|\nabla \varphi\|^2 \right)g_{ij}$

Beware that here the Laplacian $\triangle$ is minus the trace of the Hessian on functions,

$\triangle f = -\nabla^i\partial_i f$

Thus the operator $-\triangle$ is elliptic because the metric $g$ is Riemannian.

$\tilde\triangle f = e^{-2\varphi}\left(\triangle f -(n-2)\nabla^k\varphi\nabla_kf\right)$

$\tilde R = e^{-2\varphi}\left(R + 2(n-1)\triangle\varphi - (n-2)(n-1)\|\nabla\varphi\|^2\right)$

If the dimension $n>2$, then this simplifies to

$\tilde R = e^{-2\varphi}\left[R + \frac{4(n-1)}{(n-2)}e^{-(n-2)\varphi/2}\triangle\left( e^{(n-2)\varphi/2} \right) \right]$

$\tilde W^i_{jkl} = W^i_{jkl}$

We see that the (3,1) Weyl tensor is invariant under conformal changes.

Let $\omega$ be a differential $p$-form. Let $*$ be the Hodge star, and $\delta$ the codifferential. Under a conformal change, these satisfy

$\tilde * = e^{(n-2p)\varphi}*$

$\left[\tilde\delta\omega\right](v_1 , v_2 , \ldots , v_{p-1}) = e^{-2\varphi}\left[ \delta\omega - (n-2p)\omega\left(\nabla\varphi, v_1, v_2, \ldots , v_{p-1}\right) \right]$

Conformally flat manifolds

WARNING: THE FORMULAS BELOW ARE UNCHECKED AND COULD VERY WELL BE WRONG

The setting where the metric takes the form

$g_{ij} = e^{2\varphi} \delta_{ij},$

where $\delta_{ij}$ is the standard Euclidean metric, is particularly simple. These manifolds are called conformally flat. In what follows, all the partial derivatives $\partial_{i}$ and the Laplacian $\Delta$ are with respect to the Euclidean metric.

The Christoffel symbols are

$\Gamma^k_{ij} = 0$

$\Gamma^i_{ii} = \Gamma^i_{ik} = \Gamma^i_{ki} = -\Gamma^k_{ii} = \partial_k \varphi$

for $i$, $j$, and $k$ all distinct.

In this setting, the Ricci tensor takes the form

$R_{ii} = -\Delta \varphi - n \partial_i \partial_i \varphi - (n-2)(|\nabla\varphi|^2 - (\partial_i \varphi)^2)$

$R_{ij} = \partial_i \partial_i \varphi + \partial_j \partial_j \varphi - n \partial_i \partial_j \varphi + 2(n-2) \partial_i \varphi \partial_j \varphi$

for $i$ and $j$ distinct. The scalar curvature thus is

$R = g^{ij} R_{ij} = e^{-2\varphi} \delta^{ij} R_{ij} = e^{-2\varphi} (-2n \Delta \varphi - (n-1)(n-2) |\nabla \varphi|^2).$

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