Notation

Differential geometry has been described as the study of those objects that are invariant under changes in notation.

Subscripts:

Subscripts do not denote partial differentiation!

Einstein summation convention:

Repeated indices are summed unless explicitly stated otherwise!

Points in $\RR^3$ in rectangular coordinates:

$p=(p_1,p_2,p_3)$

Rectangular coordinates in $\RR^3$:

$x$, $y$, $z$ (also written $x_i$)

Tangent vectors on $\RR^3$:

$v_p$ (also written $\vv_p$) points from $p\in\RR^3$ to $p+v\in\RR^3$

Standard basis:

$U_1(p) = (1,0,0)_p, U_2(p) = (0,1,0)_p, U_3(p) = (0,0,1)_p$

Vector fields on $\RR^3$, in standard basis:

$\vv = v_i U_i = v_i\uu_i = v_x\xhat + v_y\yhat + v_z\zhat$

Action of vector fields on functions:

$\vv[f] = v_i\Partial{f}{x_i}$ (also written $\nabla_\vv f$)
$\vv[x] = v_x$, $\vv[y] = v_y$, $\vv[z] = v_z$

Curves in $\RR^3$:

$\alpha(u) = (\alpha_1(u),\alpha_2(u),\alpha_3(u))$
$x=\alpha_1(u)$, $y=\alpha_2(u)$, $z=\alpha_3(u)$

Velocity, acceleration, jerk:

$\vv = \left(\frac{d\alpha}{du}\right)_{\alpha(u)}$, $\aa = \frac{d\vv}{du} = \left(\frac{d^2\alpha}{du^2}\right)_{\alpha(u)}$, $\jj = \frac{d\aa}{du} = \left(\frac{d^3\alpha}{du^3}\right)_{\alpha(u)}$

Speed:

$v = \left|\vv\right| = \left\Vert\vv\right\Vert = \sqrt{\vv\cdot\vv}$

Arclength and unit-speed curves:

$s = \int v\,du = h(u)$, $\alpha(u) = \beta\bigl(h(u)\bigr)$ $\Longrightarrow \left|\frac{d\beta}{ds}\right| = \frac{\left|{d\alpha}/{du}\right|}{{ds}/{du}} = 1$

Frenet frame:

$\{\TT,\NN,\BB\}$, defined by $\vv = v\TT$, $\aa = \frac{dv}{du}\TT + \kappa v^2\NN$ ($\kappa\ge0$), $\BB = \TT\times\NN$

Frenet formulas:

$\frac{d\TT}{du} = \kappa v\NN$, $\frac{d\NN}{du} = -\kappa v\TT + \tau v\BB$, $\frac{d\BB}{du} = -\tau v\NN$,

Curvature and torsion:

$\kappa = \frac{\left|\vv\times\aa\right|}{v^3}$, $\tau = \frac{(\vv\times\aa)\cdot\jj}{\kappa^2v^6}$

1-forms on $\RR^3$ in standard basis:

$\psi = \psi_i dx_i = \psi_x dx + \psi_y dy + \psi_z dz$
$df = \Partial{f}{x}dx + \Partial{f}{y}dy + \Partial{f}{z}dz$

Action of 1-forms on vector fields:

$df(\vv) = \vv[f] = v_i\Partial{f}{x_i}$
$dx(\vv) = v_x$, $dy(\vv) = v_y$, $dz(\vv) = v_z$

Exterior product:

$dy\wedge dx = -dx\wedge dy$, etc.

$p$-form bases on $\RR^3$:

0-forms: $\{1\}$
1-forms: $\{dx,dy,dz\}$
2-forms: $\{dy\wedge dz,dz\wedge dx,dy\wedge dz\}$
3-forms: $\{dx\wedge dy\wedge dz\}$

Hodge dual:

${*}1=dx\wedge dy\wedge dz$, ${*}dx=dy\wedge dz$, etc.
($*$ pairs basis elements in the order shown)
${*}{*}=1$

Exterior derivative:

$d(f\,dx)=df\wedge dx$, etc.
$d(d\psi) = 0$

Product rule:

$d(\psi\wedge\eta) = d\psi\wedge\eta\pm\psi\wedge d\eta$
(sign is $(-1)^p$, where $\psi$ is a $p$-form)

Frame field:

$\ee_i\cdot\ee_j = \delta_{ij}$ (Kronecker delta)
$\ee_i = a_{ij}\uu_j$ ($\Longrightarrow \uu_j = a_{kj}\ee_k$)

Attitude matrix:

$A = \left(a_{ij}\right)$, $A^T = A^{-1}$

Covariant derivative (action of vector fields on vector fields):

$\ww=w_i\uu_i \Longrightarrow \nabla_\vv \ww = \vv[w_i]\uu_i$

Product rule:

$\nabla_\vv(f\ww) = \left(\nabla_\vv f\right)\ww + f \,\nabla_\vv \ww = \vv[f] \,\ww + f \,\nabla_\vv \ww$

Connection 1-forms:

$\nabla_\vv \ee_i = \omega_{ij}(\vv) \,\ee_j$ (most authors insert a minus sign)
$\Longrightarrow \omega_{ij}(\vv) = \nabla_\vv \ee_i \cdot \ee_j = -\omega_{ji}(\vv)$

Matrix form:

$\nabla_\vv\ee_i = \nabla_\vv(a_{ij}\uu_j) = \vv[a_{ij}]\uu_j = da_{ij}(\vv) a_{kj}\ee_k$
$\Longrightarrow \left(\omega_{ij}\right) = dA\,A^T$

Dual basis (1-forms):

$\sigma_i(\ee_j) = \delta_{ij}$
$\ee_j = a_{jk}\uu_k \Longrightarrow \sigma_i = a_{ik}dx_k$

Structure Equations:

$d\sigma_i = \omega_{ij}\wedge\sigma_j$
$d\omega_{ij} = \omega_{ik}\wedge\omega_{kj}$

Matrix Proof:

$\sigma=\left(\sigma_i\right)$, $d\xi=\left(dx_i\right)$, $\omega=\left(\omega_{ij}\right)$
$\sigma = A \,d\xi \Longrightarrow d\sigma = dA\wedge d\xi = dA\,A^T \wedge A\,d\xi = \omega \wedge \sigma$
$\omega = dA\,A^T \Longrightarrow d\omega = -dA\wedge dA^T = -dA\,A^T \wedge A\,dA^T = -dA\,A^T \wedge (dA\,A^T)^T = -\omega \wedge \omega^T = \omega\wedge\omega$