Tensors and Index Notation

Every paper on the Ricci flow is written in index notation: $\partial_tg_{ij} = -2R_{ij}$, $\Delta R = g^{ij}\nabla_i\nabla_jR$, $|\operatorname{Rm}|^2 = R_{ijkl}R^{ijkl}$. The notation is compact and powerful, and it is almost never explained in the papers, which assume it. This chapter explains it once, completely. A tensor is a multilinear function of tangent vectors and covectors (8A.3 Tangent Vectors and Bundles); in coordinates it is an array of components with upper and lower indices, transforming in a definite way when the coordinates change. The summation convention sums over each index that appears once up and once down. A metric lets you raise and lower indices, and contraction of an upper with a lower index produces quantities that don't depend on the coordinates at all.

The chapter ends with the guide's notation table and a drill of ten index computations, each of which reappears in the evolution equations of 11A.2 How Curvature Evolves. The last one, the first line of 11A.2 How Curvature Evolves, is the rehearsal.

By the end of this chapter you will be able to:

• define tensors of type $(k, l)$, tensor fields, symmetric and alternating tensors, and contractions;
• read and write expressions in index notation with the summation convention;
• use a metric to raise and lower indices, take traces and compute norms of tensors;
• state the transformation law of a tensor's components;
• use the guide's conventions for curvature, and check them on the round sphere.

7.1The stress tensor

7.2Tensors

Let $V$ be a real vector space of dimension $n$ with dual $V^*$. A tensor of type $(k, l)$ on $V$ is a multilinear map

A $(0, 1)$-tensor is a covector, a $(1, 0)$-tensor a vector (via $V \cong V^{**}$), a $(0, 2)$-tensor a bilinear form, and a $(1, 1)$-tensor a linear map $V \to V$. Given a basis $e_i$ of $V$ and the dual basis $e^i$ of $V^*$ ($e^i(e_j) = \delta^i_j$), $T$ is determined by its components

upper indices for the covector slots, lower indices for the vector slots: $n^{k+l}$ numbers.

On a manifold, tensors at each point $T_pM$ form the tensor bundles, and a tensor field is a smooth section: in coordinates, components $T^{i\dots}{}_{j\dots}(x)$ that are smooth functions, with respect to the bases $\partial_i$ of $T_pM$ and $dx^i$ of $T_p^*M$. A Riemannian metric $g$ is a symmetric positive definite $(0, 2)$-tensor field with components $g_{ij}$; the Ricci tensor $R_{ij}$ is a symmetric $(0, 2)$-tensor field; the curvature tensor $R_{ijkl}$ is a $(0, 4)$-tensor field (9A.4 Curvature and What It Means).

The summation convention. In a product, an index that appears once as a superscript and once as a subscript is summed from $1$ to $n$: $v^i\omega_i = \sum_iv^i\omega_i$, $A^i{}_jv^j$ is the vector $Av$. An index appearing once is free, and an equation must have the same free indices, in the same positions, on both sides. An index appearing twice in the same position, as in $v_iv_i$, is a warning sign: without a metric it is meaningless.

The transformation law. Under a change of coordinates, each upper index transforms with $\frac{\partial\tilde x}{\partial x}$ and each lower index with $\frac{\partial x}{\partial\tilde x}$ (8A.3 Tangent Vectors and Bundles):

and similarly for more indices. This is how one recognises that an array of functions defined in each chart is a tensor field.

Symmetric and alternating tensors. A covariant tensor is symmetric if it is unchanged by swapping any two arguments ($g_{ij} = g_{ji}$), and alternating if it changes sign ($\omega_{ij} = -\omega_{ji}$). Alternating tensors are the differential forms of 8A.8 Differential Forms and Stokes’ Theorem. The curvature tensor has a mixture of symmetries, $R_{ijkl} = -R_{jikl} = -R_{ijlk} = R_{klij}$ (9A.4 Curvature and What It Means).

Contraction. Summing an upper index against a lower one reduces the type from $(k, l)$ to $(k - 1, l - 1)$ and gives a tensor again, independent of coordinates: the trace $A^i{}_i$ of a $(1, 1)$-tensor is the trace of the linear map. Contraction is how invariants are built.

7.3The metric raises and lowers

A metric $g_{ij}$ is an invertible matrix at each point; its inverse is written $g^{ij}$:

It is a symmetric $(2, 0)$-tensor. Together they identify vectors and covectors:

so one writes the same letter with the index in either position. The gradient of a function is $df$ with its index raised, $\nabla^if = g^{ij}\partial_jf$ (8A.3 Tangent Vectors and Bundles). Raising both indices of the Ricci tensor gives $R^{ij} = g^{ik}g^{jl}R_{kl}$.

With raising and lowering, any two indices can be contracted. The trace of a $(0, 2)$-tensor is $\operatorname{tr}_gh = g^{ij}h_{ij}$; for example the scalar curvature is $R = g^{ij}R_{ij}$, and $\operatorname{tr}_gg = g^{ij}g_{ij} = \delta^i_i = n$. The inner product of two tensors of the same type contracts all indices pairwise,

and in an orthonormal frame ($g_{ij} = \delta_{ij}$) it is the sum of squares of the components. So $|g|^2 = n$, and $|\operatorname{Ric}|^2 = R_{ij}R^{ij}$ is the sum of the squares of the eigenvalues of the Ricci tensor.

7.4The guide's conventions

Different books use different conventions for curvature, differing by signs and the order of indices, and quoting a formula from one book into another is a classic source of errors. This guide uses one set throughout, following John Lee's Introduction to Riemannian Manifolds; Book 9A defines everything in it. Whatever the source, check a formula on the round unit sphere, where the answers are fixed:

quantity	definition in this guide	on the round unit $S^n$
metric, inverse	$g_{ij} = g(\partial_i, \partial_j)$, $g^{ij}g_{jk} = \delta^i_k$
curvature endomorphism	$R(X, Y)Z = \nabla_X\nabla_YZ - \nabla_Y\nabla_XZ - \nabla_{[X, Y]}Z$
curvature tensor	$\operatorname{Rm}(X, Y, Z, W) = g(R(X, Y)Z, W)$, $R_{ijkl} = \operatorname{Rm}(\partial_i, \partial_j, \partial_k, \partial_l)$	$R_{ijkl} = g_{il}g_{jk} - g_{ik}g_{jl}$
sectional curvature	$K(X, Y) = \frac{\operatorname{Rm}(X, Y, Y, X)}{\lvert X\rvert^2\lvert Y\rvert^2 - g(X, Y)^2}$	$K \equiv 1$
Ricci tensor	$R_{ij} = R_{kij}{}^k = g^{kl}R_{kijl}$	$R_{ij} = (n - 1)g_{ij}$
scalar curvature	$R = g^{ij}R_{ij}$	$R = n(n - 1)$
Hessian, Laplacian	$\nabla_i\nabla_jf$, $\Delta f = g^{ij}\nabla_i\nabla_jf$	$\Delta \leq 0$ (the "analyst's" sign)
Ricci flow	$\partial_tg_{ij} = -2R_{ij}$	$g(t) = (1 - 2(n - 1)t)g_0$

The Laplacian here is $\sum\partial_i^2$ in Euclidean space, with non-positive spectrum (4A.7 Compact Operators and Spectra), so the heat equation is $\partial_tu = \Delta u$. Some geometry books use the opposite sign; Ricci flow papers almost always use this one.

7.5Ten index computations

Each of the following takes a line or two and is used later; work through them with the summation convention, and check the free indices on both sides.

1. $\delta^i_i = n$, and $g^{ij}g_{ij} = n$.
2. Raising then lowering returns the original: $g_{ij}(g^{jk}\omega_k) = \delta_i^k\omega_k = \omega_i$.
3. Contraction is coordinate-independent: under the transformation law the Jacobians in $T^i{}_i$ cancel.
4. The trace-free part of a symmetric $h_{ij}$ is $\mathring h_{ij} = h_{ij} - \frac{\operatorname{tr}h}{n}g_{ij}$, with $\operatorname{tr}\mathring h = 0$ and $|\mathring h|^2 = |h|^2 - \frac{(\operatorname{tr}h)^2}{n}$.
5. Hence $|h|^2 \geq \frac{(\operatorname{tr}h)^2}{n}$, with equality iff $h$ is a multiple of $g$. For the Ricci tensor: $|\operatorname{Ric}|^2 \geq \frac{R^2}{n}$, the inequality behind the scalar curvature estimate of 6A.4 Maximum Principles.
6. The inner product is symmetric and positive: $|T|^2 \geq 0$, with equality iff $T = 0$ (compute in an orthonormal frame).
7. The derivative of the inverse metric. Differentiating $g^{ij}g_{jk} = \delta^i_k$: $\partial_tg^{ij}\,g_{jk} + g^{ij}\partial_tg_{jk} = 0$, so $\partial_tg^{ij} = -g^{ip}g^{jq}\partial_tg_{pq}$.
8. The derivative of the determinant. $\partial_t\log\det(g_{ij}) = g^{ij}\partial_tg_{ij}$ (Jacobi's formula, Exercise 7.4). So the volume element $dV = \sqrt{\det g}\,dx$ changes by $\partial_tdV = \frac12g^{ij}\partial_tg_{ij}\,dV$.
9. Under Ricci flow $\partial_tg_{ij} = -2R_{ij}$, item 8 gives $\partial_tdV = -R\,dV$: volume shrinks where the scalar curvature is positive, and the total volume of a closed manifold evolves by $\frac{d}{dt}\operatorname{Vol} = -\int R\,dV$.
10. Contracting a symmetric with an antisymmetric tensor gives zero: if $S_{ij} = S_{ji}$ and $A^{ij} = -A^{ji}$, then $S_{ij}A^{ij} = 0$ (relabel $i \leftrightarrow j$).

7.6History

Cauchy introduced the stress tensor in 1822–27. The calculus of components transforming by Jacobians, the absolute differential calculus, was developed by Gregorio Ricci-Curbastro from 1884 and set out with Tullio Levi-Civita in 1900; Einstein, who learned it from Marcel Grossmann, introduced the summation convention in 1916 in his foundation of general relativity, and the subject became known as tensor calculus. The coordinate-free view of tensors as multilinear maps became standard in the twentieth century. Diffusion tensor imaging dates from 1994 and tractography from 1999.

7.7Exercises

Solution

$(g^{ij})$ is the inverse matrix of $(g_{ij})$, unique and symmetric because $g$ is. If $\tilde g_{ab} = J^i{}_aJ^j{}_bg_{ij}$ with $J = \partial x/\partial\tilde x$, then the inverse is $\tilde g^{ab} = (J^{-1})^a{}_i(J^{-1})^b{}_jg^{ij}$, the $(2, 0)$ law. Polar: $g = \operatorname{diag}(1, r^2)$, $g^{-1} = \operatorname{diag}(1, r^{-2})$.

Solution

Uniaxial: trace $\sigma_0$, $s = \operatorname{diag}(\frac23, -\frac13, -\frac13)\sigma_0$, $s_{ij}s_{ij} = \frac69\sigma_0^2 = \frac23\sigma_0^2$, so $\sigma_{\mathrm{vM}} = \sqrt{\frac32\cdot\frac23}|\sigma_0| = |\sigma_0|$. Shear: trace $0$, $s = \sigma$, $s_{ij}s_{ij} = 2\tau^2$, $\sigma_{\mathrm{vM}} = \sqrt3|\tau|$.

Solution

$g^{kl}R_{kijl} = g^{kl}(g_{kl}g_{ij} - g_{kj}g_{il}) = ng_{ij} - \delta^l_jg_{il} = ng_{ij} - g_{ij}$. Then $R = g^{ij}(n - 1)g_{ij} = n(n - 1)$. For orthonormal $X = e_1$, $Y = e_2$: $\operatorname{Rm}(e_1, e_2, e_2, e_1) = g_{11}g_{22} - g_{12}g_{21} = 1$.

Solution

(a) $\partial_tg^{ij} = -g^{ip}g^{jq}(-2R_{pq}) = 2R^{ij}$. (b) $\partial_tdV = \frac12g^{ij}(-2R_{ij})dV = -R\,dV$. (c) With $\lambda = 1 - 2(n - 1)t$: $g^{ij} = \lambda^{-1}g_0^{ij}$, so $\partial_tg^{ij} = 2(n - 1)\lambda^{-2}g_0^{ij}$; and $R_{pq} = (n - 1)g_{0,pq}$ (scale-invariant), so $R^{ij} = g^{ip}g^{jq}R_{pq} = (n - 1)\lambda^{-2}g_0^{ij}$. Volume: $dV = \lambda^{n/2}dV_0$, $\partial_tdV = \frac n2\lambda^{n/2-1}(-2(n - 1))dV_0 = -\frac{n(n - 1)}{\lambda}dV$, and $R = \frac{n(n - 1)}{\lambda}$.