Differential Geometry Cheatsheet

This appendix collects the differential-geometry formulas used repeatedly in the course. It is a working reference, not a replacement for a general relativity textbook. The emphasis is on the conventions needed for AdS geometry, holographic renormalization, Brown–York stress tensors, and extremal surfaces.

Boundary data for a hypersurface $\Sigma$ embedded in a bulk spacetime $M$: unit normal $n^a$, induced metric $\gamma_{ab}=g_{ab}-\sigma n_an_b$, and extrinsic curvature $K_{ab}=\gamma_a{}^c\gamma_b{}^d\nabla_c n_d$.

Metric and index basics

The metric $g_{ab}$ lowers indices and its inverse $g^{ab}$ raises them:

$$V_a=g_{ab}V^b, \qquad V^a=g^{ab}V_b, \qquad g^{ac}g_{cb}=\delta^a{}_b.$$

The invariant volume element is

$$d^{D}x\sqrt{|g|}, \qquad D=d+1.$$

For Lorentzian metrics with mostly-plus signature, $g<0$, so $\sqrt{|g|}=\sqrt{-g}$. For Euclidean metrics, $g>0$, so $\sqrt{|g|}=\sqrt{g}$.

Useful variations are

$$\delta\sqrt{|g|} =\frac12\sqrt{|g|}\,g^{ab}\delta g_{ab} =-\frac12\sqrt{|g|}\,g_{ab}\delta g^{ab},$$

and

$$\delta g^{ab}=-g^{ac}g^{bd}\delta g_{cd}.$$

Levi-Civita connection

The Levi-Civita connection is torsion-free and metric-compatible:

$$\nabla_ag_{bc}=0, \qquad \Gamma^a{}_{bc}=\Gamma^a{}_{cb}.$$

In coordinates,

$$\Gamma^a{}_{bc} = \frac12g^{ad} \left( \partial_bg_{cd} +\partial_cg_{bd} -\partial_dg_{bc} \right).$$

For a scalar,

$$\nabla_a\phi=\partial_a\phi.$$

For a vector,

$$\nabla_aV^b=\partial_aV^b+\Gamma^b{}_{ac}V^c.$$

For a covector,

$$\nabla_a\omega_b=\partial_a\omega_b-\Gamma^c{}_{ab}\omega_c.$$

The scalar Laplacian is

$$\nabla^2\phi = \frac{1}{\sqrt{|g|}}\partial_a \left( \sqrt{|g|}g^{ab}\partial_b\phi \right).$$

In Poincaré AdS$_{d+1}$,

$$ds^2=\frac{L^2}{z^2}\left(dz^2+\eta_{ij}dx^idx^j\right),$$

this becomes

$$\nabla^2\phi = \frac{z^2}{L^2} \left( \partial_z^2\phi -\frac{d-1}{z}\partial_z\phi +\Box_{\partial}\phi \right),$$

where $\Box_{\partial}=\eta^{ij}\partial_i\partial_j$ in Lorentzian signature.

Curvature conventions

The course uses

$$[\nabla_a,\nabla_b]V^c = R^c{}_{dab}V^d.$$

In coordinates,

$$R^a{}_{bcd} = \partial_c\Gamma^a{}_{db} -\partial_d\Gamma^a{}_{cb} +\Gamma^a{}_{ce}\Gamma^e{}_{db} -\Gamma^a{}_{de}\Gamma^e{}_{cb}.$$

The Ricci tensor and scalar are

$$R_{ab}=R^c{}_{acb}, \qquad R=g^{ab}R_{ab}.$$

The Einstein tensor is

$$G_{ab}=R_{ab}-\frac12Rg_{ab}.$$

The vacuum Einstein equation with cosmological constant is

$$G_{ab}+\Lambda g_{ab}=0.$$

For AdS$_{d+1}$,

$$R_{abcd} = -\frac{1}{L^2} \left(g_{ac}g_{bd}-g_{ad}g_{bc}\right),$$

so

$$R_{ab}=-\frac{d}{L^2}g_{ab}, \qquad R=-\frac{d(d+1)}{L^2}, \qquad \Lambda=-\frac{d(d-1)}{2L^2}.$$

Maximally symmetric spaces

A $D$-dimensional maximally symmetric space has curvature

$$R_{abcd} =\frac{R}{D(D-1)} \left(g_{ac}g_{bd}-g_{ad}g_{bc}\right).$$

For a sphere of radius $L$,

$$R_{ab}=\frac{D-1}{L^2}g_{ab}, \qquad R=\frac{D(D-1)}{L^2}.$$

For AdS$_D$,

$$R_{ab}=-\frac{D-1}{L^2}g_{ab}, \qquad R=-\frac{D(D-1)}{L^2}.$$

This sign is one of the easiest ways to catch curvature-convention mistakes.

Hypersurfaces and induced metric

Let $\Sigma$ be a hypersurface with unit normal $n^a$. Define

$$\sigma=n^an_a,$$

so $\sigma=+1$ for a spacelike normal and $\sigma=-1$ for a timelike normal.

The induced metric is

$$\gamma_{ab}=g_{ab}-\sigma n_an_b.$$

It obeys

$$\gamma_{ab}n^b=0.$$

The corresponding projector is

$$\gamma^a{}_b=\delta^a{}_b-\sigma n^a n_b.$$

For tensors intrinsic to $\Sigma$, the induced covariant derivative is

$$D_aT^{b\cdots}{}_{c\cdots} = \gamma_a{}^{a'}\gamma^b{}_{b'}\cdots\gamma_c{}^{c'}\cdots \nabla_{a'}T^{b'\cdots}{}_{c'\cdots}.$$

Extrinsic curvature

The extrinsic curvature is

$$K_{ab} = \gamma_a{}^c\gamma_b{}^d\nabla_c n_d.$$

For hypersurface-orthogonal $n^a$,

$$K_{ab}=\frac12\mathcal L_n\gamma_{ab}.$$

The trace is

$$K=\gamma^{ab}K_{ab}.$$

For a radial foliation with vanishing shift,

$$ds^2=N(r,x)^2dr^2+\gamma_{ij}(r,x)dx^idx^j,$$

and normal along increasing $r$,

$$K_{ij}=\frac{1}{2N}\partial_r\gamma_{ij}.$$

If the outward normal points toward decreasing $r$, the sign flips.

Poincaré-AdS cutoff surface

For

$$ds^2=\frac{L^2}{z^2}\left(dz^2+g_{ij}(z,x)dx^idx^j\right),$$

the induced metric on $z=\epsilon$ is

$$\gamma_{ij}=\frac{L^2}{\epsilon^2}g_{ij}(\epsilon,x).$$

For the regulated region $z\ge \epsilon$, the outward unit normal is

$$n=-\frac{z}{L}\partial_z.$$

Therefore

$$K_{ij} =-\frac{z}{2L}\partial_z\gamma_{ij}.$$

For pure AdS, $g_{ij}=\eta_{ij}$ and hence

$$K_{ij}=\frac1L\gamma_{ij}, \qquad K=\frac dL.$$

Gauss–Codazzi equations

Intrinsic curvature on $\Sigma$ is related to bulk curvature by the Gauss equation:

$$\mathcal R_{abcd} = \gamma_a{}^e\gamma_b{}^f\gamma_c{}^g\gamma_d{}^hR_{efgh} + \sigma\left(K_{ac}K_{bd}-K_{ad}K_{bc}\right).$$

The Codazzi equation is

$$D_aK^a{}_b-D_bK = \gamma_b{}^cR_{cd}n^d.$$

These equations are the geometric origin of many radial constraint equations in holography. In particular, the momentum constraint becomes the boundary stress-tensor conservation Ward identity.

Gibbons–Hawking–York term

The Einstein–Hilbert action contains second derivatives of the metric. With Dirichlet boundary conditions for the induced metric, the correct variational principle requires the Gibbons–Hawking–York term:

$$S_{\mathrm{GHY}} = \frac{1}{8\pi G}\int_{\partial M}d^dx\sqrt{|\gamma|}\,K,$$

for the sign conventions used in this course. If a reference uses the opposite normal or opposite definition of $K_{ij}$, the displayed sign changes accordingly.

The regulated gravitational action is schematically

$$S_{\mathrm{reg}} = \frac{1}{16\pi G}\int_{M_\epsilon}d^{d+1}x\sqrt{|g|}\,(R-2\Lambda) + \frac{1}{8\pi G}\int_{\Sigma_\epsilon}d^dx\sqrt{|\gamma|}\,K.$$

Holographic renormalization adds local counterterms:

$$S_{\mathrm{ren}} = \lim_{\epsilon\to0} \left(S_{\mathrm{reg}}+S_{\mathrm{ct}}\right).$$

Brown–York tensor

The unrenormalized Brown–York tensor is the response to the induced metric:

$$T^{ij}_{\mathrm{BY}} = \frac{2}{\sqrt{|\gamma|}}\frac{\delta S_{\mathrm{reg}}}{\delta\gamma_{ij}}.$$

With the conventions above, its gravitational part is

$$T^{ij}_{\mathrm{BY}} = \frac{1}{8\pi G}\left(K^{ij}-K\gamma^{ij}\right),$$

before counterterms. The renormalized holographic stress tensor is obtained from

$$\langle T^{ij}\rangle = \frac{2}{\sqrt{|g_{(0)}|}}\frac{\delta S_{\mathrm{ren}}}{\delta g_{(0)ij}},$$

with Euclidean/Lorentzian signs handled as in the notation appendix.

For asymptotically AdS spacetimes, the raw Brown–York tensor diverges as the cutoff is removed. Counterterms are not optional; they are required to obtain finite CFT observables.

Fefferman–Graham expansion

Fefferman–Graham gauge writes the metric as

$$ds^2 = \frac{L^2}{z^2} \left(dz^2+g_{ij}(z,x)dx^idx^j\right).$$

The near-boundary expansion has the structure

$$g_{ij}(z,x) = g_{(0)ij} +z^2g_{(2)ij} +\cdots +z^d\left(g_{(d)ij}+h_{(d)ij}\log z^2\right) +\cdots .$$

The logarithmic term appears in even boundary dimension $d$ and is tied to the Weyl anomaly.

For pure Einstein gravity, the lower coefficients are locally determined by $g_{(0)ij}$. For example, when $d>2$,

$$g_{(2)ij} =-\frac{L^2}{d-2} \left( R_{ij}[g_{(0)}] -\frac{1}{2(d-1)}R[g_{(0)}]g_{(0)ij} \right),$$

up to the convention in which $L$ is absorbed into the definition of $z$. The coefficient $g_{(d)ij}$ contains nonlocal state data and determines $\langle T_{ij}\rangle$ after local terms are included.

Scalar near-boundary equation

For a scalar field,

$$(\nabla^2-m^2)\phi=0.$$

Near the boundary, take $\phi\sim z^\alpha$. Using the AdS Laplacian gives

$$\alpha(\alpha-d)=m^2L^2.$$

Thus

$$\alpha=\Delta \quad\text{or}\quad \alpha=d-\Delta,$$

where

$$\Delta(\Delta-d)=m^2L^2.$$

This is the geometric core of the scalar mass-dimension relation.

Differential forms

A $p$-form is

$$\omega=\frac1{p!}\omega_{a_1\cdots a_p}dx^{a_1}\wedge\cdots\wedge dx^{a_p}.$$

The exterior derivative is

$$d\omega = \frac1{p!}\partial_b\omega_{a_1\cdots a_p} dx^b\wedge dx^{a_1}\wedge\cdots\wedge dx^{a_p}.$$

For a gauge field,

$$F=dA, \qquad F_{ab}=\partial_aA_b-\partial_bA_a.$$

The Maxwell action can be written as

$$S_{\mathrm{Maxwell}} =-\frac{1}{4g_F^2}\int d^{d+1}x\sqrt{-g}\,F_{ab}F^{ab} =-\frac{1}{2g_F^2}\int F\wedge *F.$$

The Maxwell equation is

$$\nabla_aF^{ab}=0$$

or, in form language,

$$d*F=0.$$

The Bianchi identity is

$$dF=0.$$

Extremal surfaces

For a codimension-two surface $X$ with induced metric $h_{\alpha\beta}$, the area functional is

$$\mathrm{Area}(X) = \int_X d^{d-1}\sigma\sqrt{h}.$$

A minimal surface extremizes this functional on a static time slice. A covariant HRT surface extremizes the spacetime area functional and has vanishing trace of the extrinsic curvature vector:

$$K^a=0.$$

For quantum extremal surfaces, the extremized functional is the generalized entropy,

$$S_{\mathrm{gen}}[X] = \frac{\mathrm{Area}(X)}{4G_N}+S_{\mathrm{bulk}}(\Sigma_X).$$

Useful variational identities

The connection variation is

$$\delta\Gamma^a{}_{bc} = \frac12g^{ad} \left( \nabla_b\delta g_{cd} +\nabla_c\delta g_{bd} -\nabla_d\delta g_{bc} \right).$$

The Ricci variation is

$$\delta R_{ab} = \nabla_c\delta\Gamma^c{}_{ab} -\nabla_b\delta\Gamma^c{}_{ac}.$$

The Einstein–Hilbert variation has the schematic form

$$\delta\left(\sqrt{|g|}R\right) = \sqrt{|g|}\,G_{ab}\delta g^{ab} +\text{boundary term}.$$

The Gibbons–Hawking–York term cancels the part of the boundary term involving normal derivatives of $\delta\gamma_{ij}$.

Common traps

Trap 1: changing the normal changes $K_{ij}$

If $n^a\to -n^a$, then

$$K_{ij}\to -K_{ij}.$$

This changes the displayed sign of the Brown–York tensor and the Gibbons–Hawking–York term. Physical answers agree only after all conventions are changed consistently.

Trap 2: $\gamma_{ij}$ is not $g_{(0)ij}$

The cutoff induced metric diverges as

$$\gamma_{ij}\sim \frac{L^2}{\epsilon^2}g_{(0)ij}.$$

The CFT source metric is the finite conformal representative $g_{(0)ij}$, not the raw induced metric $\gamma_{ij}$.

Trap 3: intrinsic and extrinsic curvature are different

The intrinsic curvature $\mathcal R_{ijkl}[\gamma]$ is built from the induced metric on the hypersurface. The extrinsic curvature $K_{ij}$ measures how the hypersurface is embedded in the bulk. Holographic counterterms use intrinsic curvature of $\gamma_{ij}$, while Brown–York momenta use extrinsic curvature.

Trap 4: Euclidean and Lorentzian signs are not cosmetic

The Euclidean action computes $e^{-I_E}$; the Lorentzian action computes $e^{iS_L}$. One-point functions and canonical momenta can differ by signs or factors of $i$ if translated carelessly.

Exercises

Exercise 1: extrinsic curvature of the Poincaré cutoff

For pure Euclidean AdS,

$$ds^2=\frac{L^2}{z^2}\left(dz^2+\delta_{ij}dx^idx^j\right),$$

show that the cutoff surface $z=\epsilon$ has

$$K_{ij}=\frac1L\gamma_{ij}$$

when the regulated region is $z\ge\epsilon$.

Solution

The induced metric is

$$\gamma_{ij}=\frac{L^2}{z^2}\delta_{ij}.$$

For the region $z\ge\epsilon$, the outward normal points toward smaller $z$:

$$n=-\frac{z}{L}\partial_z.$$

Using

$$K_{ij}=\frac12\mathcal L_n\gamma_{ij},$$

we get

$$K_{ij} =\frac12\left(-\frac{z}{L}\right)\partial_z \left(\frac{L^2}{z^2}\delta_{ij}\right) =\frac12\left(-\frac{z}{L}\right) \left(-\frac{2L^2}{z^3}\delta_{ij}\right) =\frac{L}{z^2}\delta_{ij}.$$

Since

$$\gamma_{ij}=\frac{L^2}{z^2}\delta_{ij},$$

this is

$$K_{ij}=\frac1L\gamma_{ij}.$$

Exercise 2: scalar indicial equation

Starting from

$$\nabla^2\phi = \frac{z^2}{L^2} \left( \partial_z^2\phi -\frac{d-1}{z}\partial_z\phi +\Box_{\partial}\phi \right),$$

ignore boundary derivatives near $z=0$ and set $\phi=z^\alpha$. Derive

$$\alpha(\alpha-d)=m^2L^2.$$

Solution

For $\phi=z^\alpha$,

$$\partial_z\phi=\alpha z^{\alpha-1}, \qquad \partial_z^2\phi=\alpha(\alpha-1)z^{\alpha-2}.$$

Dropping boundary derivatives,

$$\nabla^2\phi = \frac{z^2}{L^2} \left[ \alpha(\alpha-1)z^{\alpha-2} -(d-1)\alpha z^{\alpha-2} \right] = \frac{\alpha(\alpha-d)}{L^2}z^\alpha.$$

The equation $(\nabla^2-m^2)\phi=0$ gives

$$\frac{\alpha(\alpha-d)}{L^2}=m^2,$$

or

$$\alpha(\alpha-d)=m^2L^2.$$

Exercise 3: AdS solves Einstein’s equation

Using

$$R_{ab}=-\frac{d}{L^2}g_{ab}, \qquad R=-\frac{d(d+1)}{L^2},$$

show that AdS$_{d+1}$ solves

$$R_{ab}-\frac12Rg_{ab}+\Lambda g_{ab}=0$$

with

$$\Lambda=-\frac{d(d-1)}{2L^2}.$$

Solution

Substitute the curvature data:

$$R_{ab}-\frac12Rg_{ab} = -\frac{d}{L^2}g_{ab} +\frac12\frac{d(d+1)}{L^2}g_{ab}.$$

Thus

$$R_{ab}-\frac12Rg_{ab} = \frac{d(d-1)}{2L^2}g_{ab}.$$

The Einstein equation is satisfied if

$$\frac{d(d-1)}{2L^2}g_{ab}+\Lambda g_{ab}=0,$$

so

$$\Lambda=-\frac{d(d-1)}{2L^2}.$$

Further reading

• R. M. Wald, General Relativity, for a clean modern treatment of curvature, variational principles, and causal structure.
• J. D. Brown and J. W. York, Quasilocal Energy and Conserved Charges Derived from the Gravitational Action.
• V. Balasubramanian and P. Kraus, A Stress Tensor for Anti-de Sitter Gravity.
• S. de Haro, S. N. Solodukhin, and K. Skenderis, Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence.
• K. Skenderis, Lecture Notes on Holographic Renormalization.