List of formulas in Riemannian geometry

This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwhise.

Christoffel symbols, covariant derivative

In a smooth coordinate chart, the Christoffel symbols of the first kind are given by

:$\Gamma_=\frac12 \left( \frac g_ +\frac g_ -\frac g_ \right) =\frac12 \left( g_ + g_ - g_ \right) \,,$

and the Christoffel symbols of the second kind by

:$\begin \Gamma^m_ &= g^\Gamma_\\ &=\frac\, g^ \left( \frac g_ +\frac g_ -\frac g_ \right) =\frac\, g^ \left( g_ + g_ - g_ \right) \,. \end$

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

:$\delta^i_j = g^g_$

and thus

:$n = \delta^i_i = g^i_i = g^g_$

is the dimension of the manifold.

Christoffel symbols satisfy the symmetry relations

:$\Gamma_ = \Gamma_$ or, respectively, $\Gamma^i_=\Gamma^i_$,

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

The contracting relations on the Christoffel symbols are given by

:$\Gamma^i_=\frac g^\frac=\frac \frac = \frac$

and

:$g^\Gamma^i_=\frac \;\frac$

where , ''g'', is the absolute value of the determinant of the metric tensor $g_$. 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:

:$v^i _=(\nabla_j v)^i=\frac+\Gamma^i_v^k$

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

:$v_=(\nabla_j v)_i=\frac-\Gamma^k_ v_k$

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

:$v^_=\nabla_k v^=\frac +\Gamma^i_v^+\Gamma^j_v^$

and likewise for tensors with more indices.

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

:$\nabla_i \phi=\phi_=\phi_=\frac$

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

:$(\nabla_k g)_ = 0, \quad (\nabla_k g)^ = 0$
as well as the covariant derivatives of the metric's determinant (and volume element)
:$\nabla_k \sqrt=0$
The geodesic $X(t)$ starting at the origin with initial speed $v^i$ has Taylor expansion in the chart:

:$X(t)^i=tv^i-\frac\Gamma^i_v^jv^k+O(t^3)$

Curvature tensors

Definitions

(3,1)

Riemann curvature tensor

* $^l=\frac-\frac+ \big(\Gamma_^p\Gamma_^l-\Gamma_^p\Gamma_^l\big)$
* $R(u,v)w=\nabla_u\nabla_vw-\nabla_v\nabla_uw-\nabla_w$

(3,1)

Riemann curvature tensor

* $=\frac-\frac+ \big(\Gamma_^i\Gamma_^p-\Gamma_^i\Gamma_^p\big)$

Ricci curvature

* $R_=^j$
* $\operatorname(v,w)=\operatorname(u\mapsto R(u,v)w)$

Scalar curvature

* $R= g^R_$
* $R=\operatorname_g\operatorname$

Traceless Ricci tensor

* $Q_=R_-\fracRg_$
* $Q(u,v)=\operatorname(u,v)-\fracRg(u,v)$

(4,0) Riemann curvature tensor

* $R_= ^pg_$
* $\operatorname(u,v,w,x)=g\big(R(u,v)w,x\big)$

(4,0)

Weyl tensor

* $W_=R_-\fracR\big(g_g_-g_g_\big)-\frac\big(Q_g_-Q_g_-Q_g_+Q_g_\big)$
* $W(u,v,w,x)=\operatorname(u,v,w,x)-\fracR\big(g(u,w)g(v,x)-g(u,x)g(v,w)\big)-\frac\big(Q(u,w)g(v,x)-Q(v,w)g(u,x)-Q(u,x)g(v,w)+Q(v,x)g(u,w)\big)$

Einstein tensor

* $G_=R_-\fracRg_$
* $G(u,v)=\operatorname(u,v)-\fracRg(u,v)$

Identities

Basic symmetries

* $^l=-^l$
* $R_=-R_=-R_=R_$
The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero:
* $W_=-W_=-W_=W_$
* $g^W_=0$
The Ricci tensor, the Einstein tensor, and the traceless Ricci tensor are symmetric 2-tensors:
* $R_=R_$
* $G_=G_$
* $Q_=Q_$

First Bianchi identity

* $R_+R_+R_=0$
* $W_+W_+W_=0$

Second Bianchi identity

* $\nabla_pR_+\nabla_iR_+\nabla_jR_=0$
* $(\nabla_u\operatorname)(v,w,x,y)+(\nabla_v\operatorname)(w,u,x,y)+(\nabla_w\operatorname)(u,v,x,y)=0$

Contracted second Bianchi identity

* $\nabla_jR_-\nabla_pR_=-\nabla^lR_$
* $(\nabla_u\operatorname)(v,w)-(\nabla_v\operatorname)(u,w)=-\operatorname_g\big((x,y)\mapsto(\nabla_x\operatorname)(u,v,w,y)\big)$

Twice-contracted second Bianchi identity

* $g^\nabla_pR_=\frac\nabla_k R$
*$\operatorname_g\operatorname=\fracdR$
Equivalently:
* $g^\nabla_pG_=0$
* $\operatorname_gG=0$

Ricci identity

If $X$ is a vector field then
: $\nabla_i\nabla_jX^k-\nabla_j\nabla_iX^k=-^kX^p,$
which is just the definition of the Riemann tensor. If $\omega$ is a one-form then
: $\nabla_i\nabla_j\omega_k-\nabla_j\nabla_i\omega_k=^p\omega_p.$
More generally, if $T$ is a (0,k)-tensor field then
: $\nabla_i\nabla_j T_-\nabla_j\nabla_iT_=^pT_+\cdots+^pT_.$

Remarks

A classical result says that $W=0$ if and only if $(M,g)$ is locally conformally flat, i.e. if and only if $M$ can be covered by smooth coordinate charts relative to which the metric tensor is of the form $g_=e^\varphi \delta_$ for some function $\varphi$ on the chart.

Gradient, divergence, Laplace–Beltrami operator

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

:$\nabla^i \phi=\phi^=g^\phi_=g^\phi_=g^\partial_k \phi=g^\frac$

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

:$\nabla_m V^m = \frac + V^k \frac = \frac \frac.$

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

:$\begin \Delta f &= \nabla_i \nabla^i f = \frac \frac\left(g^\sqrt\frac\right) \\ &= g^\frac + \frac \frac + \frac12 g^g^\frac\frac = g^\frac - g^\Gamma^l_\frac \end$

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

:$\nabla_k A^= \frac \frac.$

The Hessian of a map $\phi: M \rightarrow N$ is given by
:$\left( \nabla \left( d \phi\right) \right) _ ^\gamma= \frac- ^M \Gamma ^k_ \frac + ^N \Gamma ^_ \frac\frac.$

Kulkarni–Nomizu product

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

:$A_ = A_ \qquad \qquad B_ = B_$

Then we can multiply these in a sense to get a new covariant 4-tensor, which is often denoted $A B$. The defining formula is

$\left(A B\right)_ = A_B_ + A_B_ - A_B_ - A_B_$

Clearly, the product satisfies

:$A B = B A$

In an inertial frame

An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations $g_=\delta_$ and $\Gamma^i_=0$ (but these may not hold at other points in the frame). These coordinates are also called normal coordinates.
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''.

:$R_=\frac\left( \frac + \frac - \frac - \frac \right)$

:$R^\ell_= \frac \Gamma^\ell_-\frac\Gamma^\ell_$

Conformal change

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

:$\tilde g = e^g$

is also a Riemannian metric on $M$. We say that $\tilde g$ is (pointwise) 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 $g$.)

Levi-Civita connection

* $\widetilde_^k=\Gamma_^k+\frac\delta_j^k+\frac\delta_i^k-\fracg^g_$
* $\widetilde_XY=\nabla_XY+d\varphi(X)Y+d\varphi(Y)X-g(X,Y)\nabla \varphi$

(4,0) Riemann curvature tensor

* $\widetilde_=e^R_-e^\big(g_T_+g_T_-g_T_-g_T_\big)$ where $T_=\nabla_i\nabla_j\varphi-\nabla_i\varphi\nabla_j\varphi+\frac, d\varphi, ^2g_$
Using the Kulkarni–Nomizu product:
* $\widetilde = e^\operatorname - e^g \left( \operatorname\varphi - d\varphi\otimes d\varphi + \frac, d\varphi, ^2g \right)$

Ricci tensor

*$\widetilde_=R_-(n-2)\big(\nabla_i\nabla_j\varphi-\nabla_i\varphi\nabla_j\varphi\big)-\big(\Delta\varphi+(n-2), d\varphi, ^2\big)g_$
*$\widetilde=\operatorname-(n-2)\big(\operatorname\varphi-d\varphi\otimes d\varphi\big)-\big(\Delta\varphi+(n-2), d\varphi, ^2\big)g$

Scalar curvature

*$\widetilde=e^R-2(n-1)e^\Delta\varphi-(n-2)(n-1)e^, d\varphi, ^2$
* if $n\neq 2$ this can be written \tilde R = e^\left - \frace^\Delta\left( e^ \right) \right

Traceless Ricci tensor

* $\widetilde_-\frac\widetilde\widetilde_=R_-\fracRg_-(n-2)\big(\nabla_i\nabla_j\varphi-\nabla_i\varphi\nabla_j\varphi\big)+\frac\big(\Delta\varphi-, d\varphi, ^2\big)g_$
* $\widetilde-\frac\widetilde\widetilde=\operatorname-\fracRg-(n-2)\big(\operatorname\varphi-d\varphi\otimes d\varphi\big)+\frac\big(\Delta\varphi-, d\varphi, ^2\big)g$

(3,1) Weyl curvature

* $^l=^l$
* $\widetilde(X,Y,Z)=W(X,Y,Z)$ for any vector fields $X,Y,Z$

Volume form

* $\sqrt=e^\sqrt$
* $d\mu_=e^\,d\mu_g$

Hodge operator on p-forms

* $\widetilde_^=e^\ast_^$
* $\widetilde=e^\ast$

Codifferential on p-forms

* $\widetilde_^=e^(d^\ast)_^-(n-2p)e^\nabla^\varphi\delta_^\cdots\delta_^$
* $\widetilde=e^d^\ast-(n-2p)e^\iota_$

Laplacian on functions

* $\widetilde\Phi=e^\Big(\Delta\Phi + (n-2)g(d\varphi,d\Phi)\Big)$

Hodge Laplacian on p-forms

* $\widetilde\omega=e^\Big(\Delta^d\omega-(n-2p)d\circ \iota_\omega-(n-2p-2)\iota_\circ d\omega+2(n-2p)d\varphi\wedge\iota_\omega-2d\varphi\wedge d^\ast\omega\Big)$
The "geometer's" sign convention is used for the Hodge Laplacian here. In particular it has the opposite sign on functions as the usual Laplacian.

Second fundamental form of an immersion

Suppose $(M,g)$ is Riemannian and $F:\Sigma\to(M,g)$ is a twice-differentiable immersion. Recall that the second fundamental form is, for each $p\in M,$ a symmetric bilinear map $h_p:T_p\Sigma\times T_p\Sigma\to T_M,$ which is valued in the $g_$-orthogonal linear subspace to $dF_p(T_p\Sigma)\subset T_M.$ Then
* $\widetilde(u,v)=h(u,v)-(\nabla\varphi)^\perp g(u,v)$ for all $u,v\in T_pM$
Here $(\nabla\varphi)^\perp$ denotes the $g_$-orthogonal projection of $\nabla\varphi\in T_M$ onto the $g_$-orthogonal linear subspace to $dF_p(T_p\Sigma)\subset T_M.$

Mean curvature of an immersion

In the same setting as above (and suppose $\Sigma$ has dimension $n$), recall that the mean curvature vector is for each $p\in\Sigma$ an element $\textbf H_p\in T_M$ defined as the $g$-trace of the second fundamental form. Then
* $\widetilde=e^(\textbf H-n(\nabla\varphi)^\perp).$
Note that this transformation formula is for the mean curvature vector, and the formula for the mean curvature $H$ in the hypersurface case is
* $\widetilde=e^(H-n\langle\nabla\varphi,\eta\rangle)$
where $\eta$ is a (local) normal vector field.

Variation formulas

Let $M$ be a smooth manifold and let $g_t$ be a one-parameter family of Riemanannian or pseudo-Riemannian metrics. Suppose that it is a differentiable family in the sense that for any smooth coordinate chart, the derivatives $v_=\frac\big((g_t)_\big)$ exist and are themselves as differentiable as necessary for the following expressions to make sense. $v=\frac$ is a one-parameter family of symmetric 2-tensor fields.
* $\frac\Gamma_^k=\fracg^\Big(\nabla_i v_+\nabla_jv_-\nabla_pv_\Big).$
* $\fracR_=\frac\Big(\nabla_j\nabla_k v_+\nabla_i\nabla_lv_-\nabla_i\nabla_kv_-\nabla_j\nabla_lv_\Big)+\frac^pv_-\frac^pv_$
* $\fracR_=\frac\Big(\nabla^p\nabla_kv_+\nabla_i(\operatornamev)_k-\nabla_i\nabla_k(\operatorname_gv)-\Delta v_\Big)+\fracR_i^pv_-\frac R_i^p_k^qv_$
* $\fracR=\operatorname_g\operatorname_gv-\Delta(\operatorname_gv)-\langle v,\operatorname\rangle_g$
* $\fracd\mu_g=\frac g^v_\,d\mu_g$
* $\frac\nabla_i\nabla_j\Phi=\nabla_i\nabla_j\frac-\fracg^\Big(\nabla_i v_+\nabla_jv_-\nabla_pv_\Big)\frac$
* $\frac\Delta\Phi=-\langle v,\operatorname\Phi\rangle_g-g\Big(\operatornamev-\fracd(\operatorname_gv),d\Phi\Big)$

Principal symbol

The variation formula computations above define the principal symbol of the mapping which sends a pseudo-Riemannian metric to its Riemann tensor, Ricci tensor, or scalar curvature.
* The principal symbol of the map $g\mapsto\operatorname^g$ assigns to each $\xi\in T_p^\ast M$ a map from the space of symmetric (0,2)-tensors on $T_pM$ to the space of (0,4)-tensors on $T_pM,$ given by
: $v\mapsto \frac = -\frac12 (\xi \otimes \xi) v.$
* The principal symbol of the map $g\mapsto\operatorname^g$ assigns to each $\xi\in T_p^\ast M$ an endomorphism of the space of symmetric 2-tensors on $T_pM$ given by
: $v\mapsto v(\xi^\sharp,\cdot)\otimes\xi+\xi\otimes v(\xi^\sharp,\cdot)-(\operatorname_v)\xi\otimes\xi-, \xi, _g^2 v.$
* The principal symbol of the map $g\mapsto R^g$ assigns to each $\xi\in T_p^\ast M$ an element of the dual space to the vector space of symmetric 2-tensors on $T_pM$ by
: $v\mapsto , \xi, _g^2\operatorname_gv+v(\xi^\sharp,\xi^\sharp).$

See also

*Liouville equations

Notes

References

* Arthur L. Besse. "Einstein manifolds." Ergebnisse der Mathematik und ihrer Grenzgebiete (3) esults in Mathematics and Related Areas (3) 10. Springer-Verlag, Berlin, 1987. xii+510 pp. {{ISBN, 3-540-15279-2

formulas
Riemannian geometry formulas