Geometry notation

Difficulty:   ★★★★☆   graduate

Differential geometry has many distinct conventions for notation. I mostly follow those of Lee 2018 , Introduction to Riemannian manifolds §4, and Hawking & Ellis  §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 ; Schutz 2009 ) 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  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.