---
title: Manifolds, Vectors, and the Metric
module: Curved Spacetime
moduleNumber: 5
lessonNumber: 2
order: 502
summary: >
  A manifold is a space that looks locally like flat space, described by
  overlapping coordinate charts. Tangent vectors are directional derivatives with
  the coordinate basis vectors as partial-derivative operators; one-forms live in
  the dual space; and the metric tensor turns a coordinate line element into an
  invariant length. The 2-sphere and Rindler metrics serve as worked examples,
  including the coordinate singularities that are artefacts of the chart, not of
  the geometry.
topics: [Curved Spacetime]
draft: false
sources:
  - book: Carroll
    ref: "Lecture Notes on General Relativity, §2 — Manifolds"
  - book: Schutz
    ref: "A First Course in GR, Ch. 5 — Preface to Curvature; Ch. 6 — Curved Manifolds"
---

The equivalence principle guarantees that spacetime is Minkowskian in a small
neighbourhood of every event but says nothing about how those neighbourhoods fit
together. The mathematical object that is flat locally and possibly curved
globally is the **manifold**. This lesson assembles the apparatus general
relativity runs on: charts, tangent vectors as differential operators, one-forms,
and the metric tensor that recovers lengths, times, and angles from raw
coordinates. The
[four-vectors of special relativity](/relativity/spacetime-and-the-lorentz-group/four-vectors-and-index-notation)
reappear here as the tangent-space objects at a single event, with the new feature
that the tangent spaces at different events are distinct and must be connected
before they can be compared.

## Manifolds and coordinate charts

An $n$-dimensional **manifold** is a set that is locally homeomorphic to
$\mathbb{R}^n$: each point has a neighbourhood that can be mapped smoothly and
invertibly onto an open region of $\mathbb{R}^n$. Such a map is a **coordinate
chart**, assigning to each point in its patch a set of $n$ real numbers
$x^\mu$. A single chart rarely covers the whole manifold — the sphere is the
standard example, since no one chart can cover it without a singular point — so a
manifold is described by an **atlas** of charts whose patches overlap.

> **Definition (Manifold).** An $n$-dimensional differentiable manifold $M$ is a
> set together with an atlas of charts $\{(U_a, \phi_a)\}$, where the $U_a$ cover
> $M$, each $\phi_a : U_a \to \mathbb{R}^n$ is a bijection onto an open set, and
> on every overlap $U_a \cap U_b$ the transition map $\phi_b \circ \phi_a^{-1}$ is
> smooth. The transition maps are the coordinate transformations
> $x'^\mu = x'^\mu(x)$.

Where two charts overlap, a point carries two sets of coordinates related by an
invertible smooth transformation $x'^\mu(x)$, with a nonsingular Jacobian matrix
$\partial x'^\mu/\partial x^\nu$. The whole tensor calculus of general relativity
is the statement of how geometric objects transform between charts under these
Jacobians. Nothing physical depends on the choice of chart; coordinates are
labels, and a coordinate that misbehaves (the longitude at the pole) need not
signal anything wrong with the manifold.

$$
% caption: Two overlapping charts on a manifold assign different coordinates to
% the same points; on the overlap the transition map converts one labeling to the
% other with a smooth, invertible Jacobian.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % the manifold blob
  \draw[black, very thick] (0,0) .. controls (0.5,1.6) and (3.5,1.7) .. (4.2,0.4)
     .. controls (4.6,-0.5) and (3.0,-1.4) .. (1.8,-1.1)
     .. controls (0.6,-0.85) and (-0.4,-0.9) .. (0,0);
  \node[black, anchor=south] at (2.1,1.55) {manifold};
  % two overlapping patches
  \draw[acc, thick] (1.0,0.2) circle (0.95);
  \draw[acc, thick, dashed] (2.6,0.1) circle (0.95);
  \node[acc, anchor=south east] at (0.9,0.9) {patch a};
  \node[acc, anchor=south west] at (2.7,0.9) {patch b};
  % chart images below
  \draw[black] (-0.2,-3.4) rectangle (1.7,-2.0);
  \node[black, anchor=north] at (0.75,-3.4) {chart a};
  \draw[black] (2.3,-3.4) rectangle (4.2,-2.0);
  \node[black, anchor=north] at (3.25,-3.4) {chart b};
  \draw[->] (0.9,-0.85) -- (0.75,-1.95);
  \draw[->, dashed] (2.7,-0.9) -- (3.25,-1.95);
  \draw[<->, black] (1.8,-2.7) -- (2.2,-2.7);
  \node[black, anchor=south] at (2.0,-2.65) {transition};
\end{tikzpicture}
$$

## Tangent vectors as directional derivatives

In flat space a vector can be pictured as an arrow joining two points, but on a
curved manifold there is no such thing as a straight arrow between distant points,
and no canonical way to subtract the coordinates of separated points. The
definition of a vector must be intrinsic and local. The device is to identify a
vector at a point $p$ with the **directional derivative** it defines along
curves through $p$.

Let $x^\mu(\lambda)$ be a curve through $p$ and $f$ a smooth function on the
manifold. The rate of change of $f$ along the curve is

$$
\frac{\d f}{\d \lambda} = \frac{\d x^\mu}{\d\lambda}\,\frac{\partial f}{\partial x^\mu}
= \Big(\frac{\d x^\mu}{\d\lambda}\,\partial_\mu\Big) f,
$$

using the summation convention. The operator in parentheses acts on any function
and depends only on the curve's tangent at $p$. The **tangent vector** to the
curve is that operator,

$$
V = V^\mu \partial_\mu, \qquad V^\mu = \frac{\d x^\mu}{\d\lambda},
$$

and the set of all such operators at $p$ forms an $n$-dimensional vector space,
the **tangent space** $T_p M$. The coordinate partial derivatives
$\{\partial_\mu\} = \{\partial_1, \dots, \partial_n\}$ are a basis for it, the
**coordinate basis**. Reading a vector as a differential operator makes its
transformation law automatic: under a change of chart, the chain rule gives

$$
\partial_{\mu'} = \frac{\partial x^\nu}{\partial x'^\mu}\,\partial_\nu,
\qquad
V^{\mu'} = \frac{\partial x'^\mu}{\partial x^\nu}\,V^\nu.
$$

Components transform with the Jacobian $\partial x'/\partial x$ (contravariantly),
basis vectors with its inverse, and the vector $V = V^\mu\partial_\mu$ itself is
invariant. This reproduces the contravariant transformation of
[four-vectors](/relativity/spacetime-and-the-lorentz-group/four-vectors-and-index-notation),
now with a general Jacobian in place of the constant Lorentz matrix.

$$
% caption: The tangent space at a point p is spanned by the coordinate basis
% vectors, the partial-derivative operators along the coordinate lines; a curve
% through p has a tangent vector living in this flat space attached at p.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % curved surface patch (grid)
  \draw[black] (0,0) .. controls (1.5,0.4) and (3.0,0.3) .. (4.5,0.5);
  \draw[black] (0,1.0) .. controls (1.5,1.4) and (3.0,1.3) .. (4.5,1.5);
  \draw[black] (0.3,-0.4) .. controls (0.6,0.6) and (0.7,1.4) .. (0.9,2.0);
  \draw[black] (3.3,-0.2) .. controls (3.6,0.7) and (3.7,1.5) .. (3.9,2.1);
  % point p
  \fill[black!70] (2.2,1.0) circle (2pt);
  \node[black!70, anchor=north east] at (2.15,0.95) {p};
  % tangent plane
  \draw[acc, thick] (0.9,1.8) -- (3.9,1.4) -- (4.4,2.6) -- (1.4,3.0) -- cycle;
  \node[acc, anchor=south] at (2.9,3.5) {tangent space};
  % basis vectors
  \draw[->, very thick] (2.5,2.2) -- (3.7,2.05) node[right] {basis 1};
  \draw[->, very thick] (2.5,2.2) -- (2.35,3.05) node[left] {basis 2};
  % curve tangent
  \draw[->, black, very thick] (2.5,2.2) -- (3.25,2.9) node[right] {tangent};
\end{tikzpicture}
$$

## One-forms and the dual space

Alongside each tangent space sits its dual, the space of linear maps from vectors
to real numbers. Its elements are **one-forms** (covariant vectors). The
prototypical one-form is the **gradient** of a function, $\d f$, whose action on a
vector returns the directional derivative:

$$
\d f(V) = V^\mu \partial_\mu f = V(f).
$$

The coordinate differentials $\{\d x^\mu\}$ are the basis of the dual space, the
**dual basis**, defined by the pairing

$$
\d x^\mu(\partial_\nu) = \delta^\mu{}_\nu.
$$

A general one-form is $\omega = \omega_\mu \,\d x^\mu$ with a lower index, and its
components transform with the inverse Jacobian, opposite to a vector:

$$
\omega_{\mu'} = \frac{\partial x^\nu}{\partial x'^\mu}\,\omega_\nu.
$$

The pairing $\omega(V) = \omega_\mu V^\mu$ is a number independent of chart,
because the two Jacobians cancel. Vectors and one-forms are distinct species —
one cannot be added to the other — until a metric is supplied to convert between
them. A **tensor** of type $(k, l)$ is a multilinear map taking $k$ one-forms and
$l$ vectors to a number, with components $T^{\mu_1\cdots\mu_k}{}_{\nu_1\cdots\nu_l}$
carrying $k$ upper and $l$ lower indices, each transforming with its own Jacobian
factor. The metric is a $(0,2)$ tensor.

## The metric tensor and the line element

Coordinates alone carry no notion of length: the numbers $x^\mu$ could be any
smooth relabelling. The **metric tensor** $g_{\mu\nu}$ supplies the geometry,
assigning an invariant squared length to every infinitesimal displacement through
the **line element**

$$
\d s^2 = g_{\mu\nu}\,\d x^\mu\,\d x^\nu.
$$

The metric is a symmetric $(0,2)$ tensor, $g_{\mu\nu} = g_{\nu\mu}$, nondegenerate
(so its matrix is invertible), and in general relativity it has Lorentzian
signature $(-,+,+,+)$: at any point a basis can be chosen in which $g_{\mu\nu}$
reduces to $\eta_{\mu\nu} = \operatorname{diag}(-1,+1,+1,+1)$. This is the metric
statement of the equivalence principle — a local inertial frame is a chart in
which $g_{\mu\nu} = \eta_{\mu\nu}$ at the point and its first derivatives vanish.
The inverse metric $g^{\mu\nu}$, defined by $g^{\mu\nu}g_{\nu\rho} =
\delta^\mu{}_\rho$, raises indices, and $g_{\mu\nu}$ lowers them, so the metric is
the isomorphism between vectors and one-forms:

$$
V_\mu = g_{\mu\nu} V^\nu, \qquad \omega^\mu = g^{\mu\nu}\omega_\nu.
$$

The invariant scalar product of two vectors is $g_{\mu\nu}U^\mu V^\nu = U^\mu
V_\mu$, generalizing the Minkowski dot product to a position-dependent
$g_{\mu\nu}$. Lengths of curves, angles between vectors, volumes (through the
factor $\sqrt{\lvert\det g\rvert}$), and the classification of separations into
timelike, null, and spacelike all descend from the metric.

$$
% caption: The metric reads a physical length off a coordinate displacement. Equal
% coordinate steps on a curved surface span unequal true distances; the metric
% coefficients supply the local conversion factor at each point.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % a stretched coordinate grid (unequal spacing)
  \foreach \x in {0,1,2,3,4} {
    \draw[black] (\x,0) -- (\x,3);
  }
  \draw[black] (0,0) -- (4,0);
  \draw[black] (0,0.6) -- (4,0.6);
  \draw[black] (0,1.5) -- (4,1.5);
  \draw[black] (0,2.7) -- (4,2.7);
  % a displacement vector on the grid
  \draw[->, acc, very thick] (1,0.6) -- (2,1.5);
  \node[acc, anchor=south east] at (1.5,1.15) {coordinate step};
  % true length label
  \node[black, anchor=west] at (2.15,1.35) {true length};
  \node[black, anchor=west] at (2.15,1.0) {from metric};
  \node[black, anchor=north] at (2.0,-0.1) {equal coordinate steps, unequal distances};
\end{tikzpicture}
$$

## Worked example: the 2-sphere

The surface of a sphere of radius $a$ is the canonical curved two-dimensional
manifold. In the usual polar and azimuthal angles $(\theta, \phi)$, the induced
line element is

$$
\d s^2 = a^2\,\d\theta^2 + a^2 \sin^2\theta\,\d\phi^2,
\qquad
g_{\mu\nu} = \begin{pmatrix} a^2 & 0 \\ 0 & a^2 \sin^2\theta \end{pmatrix}.
$$

The metric is diagonal but not constant: the coefficient of $\d\phi^2$ shrinks
toward the poles, capturing that circles of constant latitude get smaller as
$\theta \to 0$ or $\pi$. Two features are worth separating. First, the
**intrinsic curvature** is real — no coordinate change can flatten the sphere,
as later lessons confirm by computing a nonzero Riemann tensor. Second, the
apparent breakdown of the coordinates at $\theta = 0, \pi$, where $\det g = a^4
\sin^2\theta$ vanishes, is a **coordinate singularity**: the poles are perfectly
ordinary points, but the chart $(\theta,\phi)$ fails there because longitude is
undefined at a pole. A second chart, rotated so its poles lie elsewhere, covers
the region smoothly. Distinguishing a coordinate singularity from a genuine one is
a recurring theme; the Schwarzschild horizon is a coordinate singularity, while
its centre is a real one.

> **Worked example.** The circumference of the latitude circle at polar angle
> $\theta$ is found by integrating the line element along constant $\theta$,
> $\d s = a\sin\theta\,\d\phi$:
> $$
> C(\theta) = \int_0^{2\pi} a\sin\theta\,\d\phi = 2\pi a \sin\theta.
> $$
> Its proper radius, measured along a meridian from the pole, is
> $r = \int_0^\theta a\,\d\theta' = a\theta$, so $C = 2\pi a\sin(r/a) < 2\pi r$
> for $r > 0$. The circumference falls short of the flat-space value $2\pi r$, and
> the deficit is the observable signature of positive curvature.

$$
% caption: On the 2-sphere the latitude circle has circumference 2 pi a sin(theta),
% less than 2 pi times its meridian radius a theta; the shortfall is the flat-space
% deficit that marks positive intrinsic curvature. The poles are coordinate, not
% geometric, singularities.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % sphere outline
  \draw[black, very thick] (0,0) circle (2);
  % equator ellipse
  \draw[black, dashed] (0,0) ellipse (2 and 0.6);
  % latitude circle upper
  \draw[acc, very thick] (0,1.3) ellipse (1.52 and 0.42);
  \node[acc, anchor=south] at (0,1.72) {latitude circle};
  % meridian arc from pole
  \draw[black, very thick] (0,2) .. controls (0.9,1.9) and (1.45,1.55) .. (1.52,1.3);
  \node[black, anchor=west] at (1.0,1.95) {meridian};
  % pole
  \fill[black!70] (0,2) circle (2pt);
  \node[black!70, anchor=south west] at (0.05,2.0) {pole};
  \node[black, anchor=north] at (0,-2.05) {radius from pole = a theta};
\end{tikzpicture}
$$

## Worked example: the Rindler chart

The uniformly accelerated frame of the previous lesson is a flat manifold in
curved-looking coordinates, the reverse situation from the sphere. Its metric,

$$
\d s^2 = -\Big(\frac{\alpha\rho}{c^2}\Big)^2 c^2\,\d\tau^2 + \d\rho^2,
$$

has a position-dependent $g_{\tau\tau}$ and vanishing $\det g$ at $\rho = 0$. That
zero is a coordinate singularity — the Rindler horizon — not a curvature
singularity, since the underlying spacetime remains Minkowski. Computing the
curvature (done systematically two lessons on) returns zero everywhere, confirming
that a nonconstant metric does not by itself imply curvature. The metric matrix
carries two kinds of information tangled together: the true geometry, and the
choice of coordinates. Separating them requires the covariant derivative and the
curvature tensor, developed next.

$$
% caption: A nonconstant metric can describe flat spacetime (Rindler) or curved
% (sphere). The metric matrix alone cannot tell them apart; only its derivatives,
% assembled into curvature, distinguish a coordinate effect from real geometry.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % two columns
  \draw[black] (-0.3,-0.4) rectangle (3.0,2.4);
  \node[black, anchor=south] at (1.35,2.4) {sphere metric};
  \node[acc, anchor=north] at (1.35,1.9) {nonconstant g};
  \node[black, anchor=north] at (1.35,1.3) {curvature nonzero};
  \node[black, anchor=north] at (1.35,0.6) {truly curved};
  \draw[black] (3.6,-0.4) rectangle (6.9,2.4);
  \node[black, anchor=south] at (5.25,2.4) {Rindler metric};
  \node[acc, anchor=north] at (5.25,1.9) {nonconstant g};
  \node[black, anchor=north] at (5.25,1.3) {curvature zero};
  \node[black, anchor=north] at (5.25,0.6) {Minkowski};
  \node[black, anchor=north] at (3.3,-0.6) {same symptom, distinct geometry};
\end{tikzpicture}
$$

The manifold supplies the arena and the metric supplies lengths and times, but
comparing vectors at different points — the operation every derivative of a
vector field requires — has no meaning yet, because the tangent spaces at
different events are distinct vector spaces. Supplying that comparison is the work
of the connection and the covariant derivative, the subject of the next lesson.

[^carroll-mfld]: **Carroll**, _Lecture Notes on General Relativity_, §2 — manifolds and atlases, tangent vectors as directional derivatives with the coordinate basis $\partial_\mu$, one-forms and the dual basis $\d x^\mu$, tensors, and the metric with its Lorentzian signature.

[^schutz-curved]: **Schutz**, _A First Course in General Relativity_, Ch. 5–6 — the transition from flat to curved manifolds, the metric and line element, the 2-sphere as a worked example, and the distinction between coordinate and genuine singularities.