Geometry notation – CMacLaurin.com

Difficulty: ★★★★☆ graduate

Differential geometry has many distinct conventions for notation. I mostly follow those of Lee 2018 book , Introduction to Riemannian manifolds §4, and Hawking & Ellis book §2.5. I use the metric signature -+++ preferred by modern relativists most of the time, and geometric units with $G = c = 1$ . I have recently grown fond of using angle brackets to signify contraction using the metric, so for vectors $\mathbf X$ and $\mathbf Y$ :

$\langle\mathbf X,\mathbf Y\rangle := g_{\mu\nu}X^\mu Y^\nu.$

Many (Hawking & Ellis §2.2; MTW book ; Schutz 2009 book ) also use this notation for a 1-form and vector: $\langle\boldsymbol\omega,\mathbf Y\rangle := \omega_\mu Y^\mu = \delta^\mu_{\hphantom\mu \nu}\omega_\mu Y^\nu$ , which may also be written $\boldsymbol\omega(\mathbf Y)$ . Presumably angle brackets are also used for a pair of 1-forms: $\langle\boldsymbol\omega,\boldsymbol\alpha\rangle := g^{\mu\nu}\omega_\mu \alpha_\nu$ .

I like the index-free notation. This is not to say components don’t have their place: “the abstract notation… is poorly suited to complex calculations; but it possesses great conceptual power.” (MTW §8.4) Hence, for a connection or derivative I use the following order of indices:

$\nabla_\mu Y^\nu,$

so the first index $\mu$ specifies the direction of differentiation, and the second (and later indices, for an arbitrary tensor) describe the quantity being differentiated. This choice is consistent with the index-free notation $\nabla_{\mathbf X}\mathbf Y$ , which has components $X^\mu\nabla_\mu Y^\nu$ incidentally. It has been called the “del convention”, as opposed to the “semi-colon convention” which would naturally suggest the opposite order $Y^\mu_{\hphantom\mu;\nu}$ (Jantzen+ 2013 book draft, Understanding spacetime splittings and their relationships §2.3.2). The $\nabla$ -convention is used by Hawking & Ellis, as well as Lee (but seemingly not for the following).

For the total covariant derivative $\nabla\mathbf Y$ , I tentatively define the first slot to be the differentiation direction, so $\nabla\mathbf Y(\mathbf X,\cdot) \equiv \nabla_{\mathbf X}\mathbf Y$ . This is used by Hawking & Ellis, but admittedly Lee and every other book I checked use the last slot instead. Anyway, with this ordering we have the product rule:

$\nabla(f\mathbf Y) = df\otimes\mathbf Y + f\nabla\mathbf Y,$

which in the $\mathbf X$ direction reduces to the more common formula $\nabla_{\mathbf X}(f\mathbf Y) = \langle df,\mathbf X\rangle\mathbf Y + f\nabla_{\mathbf X}\mathbf Y$ as required. Here $f$ is a scalar field and $df\equiv\nabla f$ its gradient 1-form, where $d$ is the exterior derivative. The contraction of $df$ with $\mathbf X$ is equivalently written $df(\mathbf X)$ , $\mathbf X(f)$ , $\partial_{\mathbf X}f$ , or $\nabla_{\mathbf X}f$ .

Lee defines the connection coefficients for a given frame field to satisfy $\nabla_{\mathbf e_i}\mathbf e_j = \Gamma_{ij}^k\mathbf e_k$ . This seems to imply the alphabetical ordering $\Gamma_{ij}^{\hphantom{ij}k}$ ; either way it makes sense to me to put the basis indices first and the component index last, because:

$\Gamma_{ij}^{\hphantom{ij}k} = (\nabla_{\mathbf e_i}\mathbf e_j)^k.$

This is consistent with our total derivative notation, in which those indices which pick out components of the “result” or answer are placed last. [It is also consistent with $\Gamma_{ij}^{\hphantom{ij}k} = \langle\nabla_{\mathbf e_i}\mathbf e_j,\mathbf e^k\rangle$ , and the first term in the (Levi-Civita) Christoffel symbols below, though these can also be made to align for most conventions. Admittedly most textbooks seem to put the “up” index first. A possible justification is that in the covariant derivative below, $\Gamma$ is part of an operator acting on two vectors, and operators are usually written on the left? Another is that for a (1,2)-tensor for example, the numbers count the upper indices first, so it feels a bit more consistent to order them that way. Of course there is no ambiguity in just writing $\Gamma_{ij}^k$ , since $k$ is distinguished by being the only upper index. But if nothing else, fixing an order is needed for my own computer algebra, where I store arrays of components with no internal record distinguishing upper from lower.]

As a reminder, the covariant derivative of a vector has the components:

$(\nabla_XY)^k = X^i\partial_iY^k + \Gamma_{ij}^{\hphantom{ij}k}X^iY^j.$

For the Levi-Civita connection, the Christoffel symbols are given in a coordinate frame by:

$\Gamma_{ijk} = \frac{1}{2}\big(\partial_i g_{jk} + \partial_j g_{ik} - \partial_k g_{ij}\big),$

and $\Gamma_{ij}^{\hphantom{ij}k} = \Gamma_{ijm}\,g^{mk}$ . For the Riemann curvature tensor, it seems every recent source defines $\mathbf R(\mathbf X,\mathbf Y)\mathbf Z = \nabla_{\mathbf X}\nabla_{\mathbf Y}\mathbf Z - \nabla_{\mathbf Y}\nabla_{\mathbf X}\mathbf Z - \nabla_{[\mathbf X,\mathbf Y]}\mathbf Z$ . Given this, Lee’s choice (§7) for coordinate components seems entirely consistent with all the above conventions:

$R_{ijk}^{\hphantom{ijk}l} = \partial_i\Gamma_{jk}^{\hphantom{jk}l} - \partial_j\Gamma_{ik}^{\hphantom{ik}l} + \Gamma_{jk}^{\hphantom{jk}m}\Gamma_{im}^{\hphantom{im}l} - \Gamma_{ik}^{\hphantom{ik}m}\Gamma_{jm}^{\hphantom{jm}l}.$

[Lee states “almost all authors” use this sign convention. The definition is equivalent to MTW §8.7 and§11.3, after comparing the $\mathbf R(\cdot,\cdot)$ definitions, and swapping the lower Christoffel symbol indices (because MTW define them this way in §8.5; also they probably only use the Levi-Civita connection, which is symmetric). In the front matter of their book MTW say their sign choice for Riemann is “convenient for coordinate-free methods”, and their Ricci tensor sign keeps eigenvalues and scalar curvature “positive for standard spheres with positive definite metrics.”] The Ricci tensor is $R_{jk} := R_{ijk}^{\hphantom{ijk}i}$ (Lee; it turns out MTW §8.7 have an equivalent definition). The scalar curvature is $g^{ij}R_{ij}$ .

The “musical isomorphism” notations $^\sharp$ and $^\flat$ raise or lower an index slot, respectively.

The wedge product of two 1-forms will likely be set as: $\boldsymbol\alpha\wedge\boldsymbol\beta := \frac{1}{2}(\boldsymbol\alpha\otimes\boldsymbol\beta-\boldsymbol\beta\otimes\boldsymbol\alpha)$ . Some imply the symmetric product simply by juxtaposing terms: $dx\,dy := \frac{1}{2}(dx\otimes dy+dy\otimes dx)$ , especially in the case of the metric.

People have different preferences on notation. This was written in April 2021, during a careful overhaul of my own conventions, especially as used in computer algebra. For the reader, in the very least they aid internal consistency within this website.