Riemannian Geometry Basics

Riemannian geometry is the study of smooth manifolds equipped with a smoothly varying inner product on each tangent space (a Riemannian metric), which lets one measure lengths, angles, areas, and curvature intrinsically. Key result: the metric determines a unique torsion-free compatible connection (Levi-Civita), whose curvature obstructs local flatness and controls geodesic divergence. Applies to general relativity, geometric analysis, comparison geometry, shape analysis. Founded by Bernhard Riemann (1854), formalized by Levi-Civita, Ricci, and Élie Cartan.

A Riemannian manifold $(M, g)$ is a smooth manifold $M$ together with a metric tensor $g$ , a smooth assignment to each point $p$ of a positive-definite symmetric bilinear form $g_{p}$ on the tangent space $T_{p} M$ . This single structure generates a rich derived hierarchy: lengths of curves, distances, geodesics, parallel transport, and curvature — all computed intrinsically, without reference to any ambient embedding. The decisive fact, often surprising to newcomers, is that although a metric "looks" like 1+2+...+n free functions, its second-order invariant content is captured by the Riemann curvature tensor, and a manifold is locally isometric to Euclidean space if and only if that curvature vanishes identically. Riemannian geometry thereby unifies the infinitesimal (the metric and connection) with the global (topology, completeness, conjugate points) through comparison and variational principles.

The Core Insight

Geometry on a manifold is not added by embedding it in $R^{N}$ ; it is generated internally by a choice of inner product on tangent spaces. From the metric $g$ alone, one canonically recovers a differentiation operator (the Levi-Civita connection $\nabla$ ) and a measure of its non-commutativity (the curvature $R$ ). Everything else — geodesics, distance, volume, sectional curvature — is downstream. The leap from Gauss's theorema egregium (curvature is intrinsic to a surface) to Riemann's program is the recognition that this intrinsic viewpoint works in any dimension and needs no embedding at all.

Formal Statement

Riemannian Metric

Let $M$ be a smooth $n$ -manifold. A Riemannian metric $g$ is a smooth section of the bundle $Sym^{2} T^{*} M$ such that for each $p \in M$ , $g_{p} : T_{p} M \times T_{p} M \to R$ is symmetric and positive-definite. In local coordinates $(x^{1}, \dots, x^{n})$ with coordinate vector fields $\partial_{i}$ ,

g = g_{ij} d x^{i} \otimes d x^{j}, g_{ij} = g (\partial_{i}, \partial_{j}) = g_{j i}, (g_{ij}) ≻ 0.

⟳Expand

g

in the basis

d x^{i} \otimes d x^{j}

T^{*} M \otimes T^{*} M

; symmetry forces

g_{ij} = g_{j i}

, and positive-definiteness of the matrix encodes

g_{p} (v, v) > 0

for

v \neq = 0

The length of a piecewise- $C^{1}$ curve $γ : [a, b] \to M$ and the induced Riemannian distance are

L (γ) = \int_{a}^{b} g_{γ (t)} (\overset{γ}{˙} (t), \overset{γ}{˙} (t)) d t, d (p, q) = γ : p \to q in f L (γ) .

⟳Integrate the infinitesimal length element

d s = g_{ij} \overset{x}{˙}^{i} \overset{x}{˙}^{j} d t

; the distance is the infimum over all admissible curves joining

p

q

, which makes

(M, d)

a metric space inducing the manifold topology.

Levi-Civita Connection

On a Riemannian manifold there is a unique affine connection $\nabla$ that is metric-compatible ( $\nabla g = 0$ ) and torsion-free ( $\nabla_{X} Y - \nabla_{Y} X = [X, Y]$ ). Its Christoffel symbols are

Γ_{ij}^{k} = \frac{1}{2} g^{k ℓ} (\partial_{i} g_{j ℓ} + \partial_{j} g_{i ℓ} - \partial_{ℓ} g_{ij}), \nabla_{\partial_{i}} \partial_{j} = Γ_{ij}^{k} \partial_{k},

where

g^{k ℓ}

is the inverse metric,

g^{k ℓ} g_{ℓ j} = δ_{j}^{k}

⟳Write the Koszul formula

2 g (\nabla_{X} Y, Z) = X g (Y, Z) + Y g (X, Z) - Z g (X, Y) + g ([X, Y], Z) - g ([X, Z], Y) - g ([Y, Z], X)

, evaluate on coordinate fields where brackets vanish, and solve for

Γ_{ij}^{k}

Fundamental Theorem of Riemannian Geometry

For any Riemannian manifold $(M, g)$ there exists a unique connection $\nabla$ that is simultaneously torsion-free and compatible with $g$ . It is given explicitly by [3].

Existence and Uniqueness via Koszul

Step 1 (Uniqueness). Assume

\nabla

is metric-compatible and torsion-free. Metric compatibility means

X g (Y, Z) = g (\nabla_{X} Y, Z) + g (Y, \nabla_{X} Z) .

⟳This is

\nabla g = 0

written out via the product rule for the connection acting on the tensor

g

Cyclically permute $X, Y, Z$ in [4], add the first two equations and subtract the third, then apply torsion-freeness $\nabla_{X} Y - \nabla_{Y} X = [X, Y]$ to eliminate antisymmetric terms. One obtains the Koszul formula

2 g (\nabla_{X} Y, Z) = X g (Y, Z) + Y g (X, Z) - Z g (X, Y) + g ([X, Y], Z) - g ([X, Z], Y) - g ([Y, Z], X) .

⟳The right side involves only

g

, the vector fields, and their Lie brackets — none of

\nabla

— so the left side determines

\nabla_{X} Y

uniquely because

g

is nondegenerate.

Step 2 (Existence). Define

\nabla

by declaring [5] to hold; nondegeneracy of

g

lets us solve for

\nabla_{X} Y

. One then verifies directly that this

\nabla

R

-bilinear,

C^{\infty} (M)

-linear in

X

, Leibniz in

Y

, torsion-free, and metric-compatible. In coordinates the formula collapses to [3].

■

A geodesic is a curve with vanishing covariant acceleration; in coordinates,

\overset{γ}{¨}^{k} + Γ_{ij}^{k} \overset{γ}{˙}^{i} \overset{γ}{˙}^{j} = 0.

⟳Set

\nabla_{\overset{γ}{˙}} \overset{γ}{˙} = 0

and expand using

\overset{γ}{˙} = \overset{γ}{˙}^{i} \partial_{i}

and [3]; the geodesic equations are the Euler–Lagrange equations of the energy functional

E (γ) = \frac{1}{2} \int g (\overset{γ}{˙}, \overset{γ}{˙}) d t

Riemann Curvature Tensor

The curvature of $\nabla$ measures the failure of second covariant derivatives to commute:

R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z .

⟳This is a

(1, 3)

-tensor (it is

C^{\infty}

-linear in all three arguments despite being built from derivatives — the bracket term cancels the non-tensorial pieces). Its components are

R^{ℓ}_{k ij} = \partial_{i} Γ_{j k}^{ℓ} - \partial_{j} Γ_{ik}^{ℓ} + Γ_{im}^{ℓ} Γ_{j k}^{m} - Γ_{j m}^{ℓ} Γ_{ik}^{m}

The sectional curvature of a 2-plane $σ = span (X, Y) \subset T_{p} M$ is

K (σ) = \frac{g ( R ( X , Y ) Y , X )}{g ( X , X ) g ( Y , Y ) - g ( X , Y ) ^{2}} .

⟳The denominator is the squared area of the parallelogram spanned by

X, Y

; the quotient is independent of the chosen basis of

σ

because

R

enjoys the symmetries

R_{ij k ℓ} = - R_{j ik ℓ} = - R_{ij ℓ k} = R_{k ℓ ij}

The Ricci and scalar curvatures are traces:

Ric (Y, Z) = tr (X \mapsto R (X, Y) Z) = R^{k}_{y k z}, S = g^{ij} Ric_{ij} .

⟳Contract the Riemann tensor on its first and third slots to obtain Ric; contract Ric with the inverse metric to obtain the scalar

S

. These traces feed directly into the Einstein equations and Bochner formulas.

Why It Holds / Derivation

Flatness ⟺ Vanishing Curvature

Claim.

(M, g)

is locally isometric to Euclidean

(R^{n}, δ)

iff

R \equiv 0

. Step 1. If

g

is locally Euclidean, then in flat coordinates

g_{ij} = δ_{ij}

, so all

Γ_{ij}^{k} = 0

by [3], hence

R = 0

by [7]. Step 2 (converse). Suppose

R \equiv 0

. Parallel transport then becomes path-independent on simply connected neighborhoods, because the holonomy around small loops is generated by

R

(Ambrose–Singer). Parallel-transport an orthonormal frame

{e_{a}}

from

p

; flatness guarantees the resulting fields are globally defined and satisfy

\nabla e_{a} = 0

. Step 3. Torsion-freeness gives

[e_{a}, e_{b}] = \nabla_{e_{a}} e_{b} - \nabla_{e_{b}} e_{a} = 0

, so the frame is coordinate-inducing (Frobenius): there are coordinates

y^{a}

with

\partial / \partial y^{a} = e_{a}

. Metric compatibility makes

g (e_{a}, e_{b}) = δ_{ab}

constant, so

g = \sum (d y^{a})^{2}

in these coordinates.

■

Geodesic spread on positively curved vs. flat space: a positive sectional curvature focuses nearby geodesics.

The separation of nearby geodesics is governed by the Jacobi equation:

\nabla_{\overset{γ}{˙}} \nabla_{\overset{γ}{˙}} J + R (J, \overset{γ}{˙}) \overset{γ}{˙} = 0.

⟳Take a variation

γ_{s} (t)

of geodesics, let

J = \partial_{s} γ_{s} ∣_{s = 0}

be the variation field, differentiate the geodesic equation

\nabla_{\partial_{t}} \partial_{t} γ = 0

s

, and commute derivatives using the curvature definition [7].

Bernhard Riemann | 1826–1866

Riemann introduced the entire framework in his 1854 Göttingen Habilitationsvortrag, "Über die Hypothesen, welche der Geometrie zu Grunde liegen," delivered before Gauss. Without modern tensor notation he posited an $n$ -dimensional "manifoldness" with an infinitesimal line element $d s = \sum g_{ij} d x^{i} d x^{j}$ and identified the obstruction to flatness — what we now call the curvature tensor — via the quadratic deviation of geodesic distances. He saw clearly that physical space need not be Euclidean and that geometry is an empirical question, anticipating general relativity by sixty years. His ideas were made computationally precise by Ricci-Curbastro and Levi-Civita (the absolute differential calculus, 1900) and parallel transport (1917).

Building Intuition

Think of a Riemannian metric as a device that tells you the ruler at each point: $g_{p} (v, v)$ is the squared length of the infinitesimal displacement $v$ . On a sphere the ruler shrinks lines of longitude near the poles; that distortion is the geometry. The Levi-Civita connection answers "how do I carry a vector along without artificially rotating it?" — it is the unique notion of straight-ahead transport that respects lengths and angles and adds no twist. Curvature then answers "if I carry a vector around a tiny loop, how much did it rotate?" — the answer is $R$ times the loop's area. Positive curvature (sphere) makes geodesics converge like meridians; negative curvature (hyperbolic plane) makes them diverge exponentially; zero curvature (plane, flat torus) keeps them parallel. Every global theorem in the subject is a careful bookkeeping of these local rotations summed over regions.

Worked Examples

Curvature of the Round Sphere $S^2_r$

Given. The sphere of radius

r

with metric in spherical coordinates

g = r^{2} d θ^{2} + r^{2} sin^{2} θ d ϕ^{2}

, so

g_{θ θ} = r^{2}

g_{ϕϕ} = r^{2} sin^{2} θ

. Step 1 — Christoffels. Using [3] with the diagonal metric:

Γ_{ϕϕ}^{θ} = - sin θ cos θ

Γ_{θ ϕ}^{ϕ} = Γ_{ϕθ}^{ϕ} = cot θ

, all others zero. Step 2 — Riemann component. Compute

R^{θ}_{ϕθ ϕ} = \partial_{θ} Γ_{ϕϕ}^{θ} - \partial_{ϕ} Γ_{θ ϕ}^{θ} + Γ_{θ m}^{θ} Γ_{ϕϕ}^{m} - Γ_{ϕ m}^{θ} Γ_{θ ϕ}^{m}

\partial_{θ} (- sin θ cos θ) = - cos 2 θ = sin^{2} θ - cos^{2} θ

, and

- Γ_{ϕϕ}^{θ} Γ_{θ ϕ}^{ϕ} = - (- sin θ cos θ) (cot θ) = cos^{2} θ

. Sum:

sin^{2} θ

. Step 3 — Sectional curvature. Using [8] with the only 2-plane,

K = \frac{g ( R ( \partial _{θ} , \partial _{ϕ} ) \partial _{ϕ} , \partial _{θ} )}{g _{θ θ} g _{ϕϕ}} = \frac{r ^{2} sin ^{2} θ \cdot R ^{θ} _{ϕθ ϕ} / sin ^{2} θ}{r ^{2} \cdot r ^{2} sin ^{2} θ}

After substitution, $g (R \dots) = g_{θ θ} R^{θ}_{ϕθ ϕ} = r^{2} sin^{2} θ$ , so

K = \frac{r ^{2} sin ^{2} θ}{r ^{4} sin ^{2} θ} = \frac{1}{r ^{2}} [units: length^{- 2}]

Constant positive curvature

1/ r^{2}

, independent of

θ, ϕ

. Scalar curvature

S = 2 K = 2/ r^{2}

Geodesics and Distance on the Hyperbolic Plane $\mathbb{H}^2$

Given. Upper half-plane

{(x, y) : y > 0}

with

g = \frac{d x ^{2} + d y ^{2}}{y ^{2}}

, so

g_{xx} = g_{y y} = y^{- 2}

. Step 1 — Christoffels. From [3]:

Γ_{x y}^{x} = Γ_{y x}^{x} = - 1/ y

Γ_{xx}^{y} = 1/ y

Γ_{y y}^{y} = - 1/ y

. Step 2 — Curvature. Computing

R^{x}_{y x y}

analogously gives sectional curvature

K \equiv - 1

everywhere (constant negative). Step 3 — Vertical geodesic distance. Along

x =

const,

γ (t) = (x_{0}, t)

, the length between heights

y_{1} < y_{2}

L = \int_{y_{1}}^{y_{2}} \frac{0 + 1}{y} d y = ln \frac{y _{2}}{y _{1}} .

For

(x_{0}, 1)

(x_{0}, e)

L = ln (e /1) = 1

. Vertical lines and semicircles centered on the

x

-axis are the geodesics. Step 4 — A numeric distance. Distance from

A = (0, 1)

B = (0, e^{2})

d (A, B) = ln \frac{e ^{2}}{1} = 2 [dimensionless hyperbolic units]

Common Misconceptions

Curvature is not the second derivative of the metric in any frame-independent naive sense. Because

\partial_{k} g_{ij}

is not a tensor, you cannot read off curvature from a single coordinate patch by inspection. At any point you can choose normal coordinates where

g_{ij} = δ_{ij}

and

\partial_{k} g_{ij} = 0

; the curvature lives in the second derivatives that cannot be killed,

g_{ij} = δ_{ij} - \frac{1}{3} R_{ik j ℓ} x^{k} x^{ℓ} + O (∣ x ∣^{3})

. Vanishing Christoffels at a point does not mean flatness.

Flat Metric in Curvilinear Coordinates

Polar coordinates on the plane give $g = d r^{2} + r^{2} d θ^{2}$ , whose Christoffel symbols are nonzero: $Γ_{θ θ}^{r} = - r, Γ_{r θ}^{θ} = 1/ r$ . A student might think "nonzero $Γ$ ⟹ curved." But computing [7] yields $R \equiv 0$ : this is just the flat Euclidean plane in disguise. Conversely, the flat torus $R^{2} / Z^{2}$ has $g = δ$ , $R \equiv 0$ , yet is globally distinct from $R^{2}$ (compact, nontrivial $π_{1}$ ). Curvature is a local invariant; flatness does not pin down the global topology.

The distance infimum in [2] need not be achieved. On an incomplete manifold (e.g. the punctured plane) there may be no minimizing geodesic between two points. The Hopf–Rinow theorem rescues us: for a connected manifold, geodesic completeness, metric completeness, and "closed bounded sets are compact" are equivalent, and any two points are then joined by a minimizing geodesic. Never assume minimizers exist without completeness.

Connections

Riemannian geometry sits at a crossroads. The Levi-Civita connection is a special case of an Affine Connection on a principal bundle, and its curvature is a component of the general Curvature of Connections formalism; the holonomy group of $\nabla$ links to Lie Groups and the Berger classification. Restricting to constant sectional curvature recovers the model geometries underlying Hyperbolic Geometry and spherical geometry. Adding a Lorentzian signature (dropping positive-definiteness) yields Pseudo-Riemannian Geometry, the arena of General Relativity, where Ric and $S$ enter the Einstein field equations. The interplay between curvature bounds and topology is the domain of Comparison Geometry (Bonnet–Myers, Rauch, Toponogov), while the analytic side — eigenvalues of the Laplace–Beltrami operator, heat kernels, Bochner technique — feeds Geometric Analysis.

Real-World Applications

General Relativity — spacetime is a 4D pseudo-Riemannian manifold; geodesics [6] are the worldlines of free-falling bodies and the Ricci tensor [9] sources gravity via $G_{μν} = 8 π T_{μν}$ .
Medical imaging / DTI — diffusion tensor imaging models water diffusion in brain tissue as a Riemannian metric; geodesics trace white-matter fiber tracts (tractography).
Shape analysis & computer vision — spaces of shapes (Kendall shape space, the manifold of SPD matrices) carry natural Riemannian metrics; geodesic distance compares anatomical shapes and covariance descriptors.
Robotics & control — configuration spaces of mechanisms are Riemannian manifolds; the kinetic-energy metric makes trajectory planning a geodesic problem, and curvature governs controllability.
Machine learning (information geometry) — the Fisher information metric turns statistical model spaces into Riemannian manifolds; natural-gradient descent moves along the metric's steepest direction.
Optimization on manifolds — Riemannian optimization (Stiefel, Grassmann manifolds) solves PCA, matrix completion, and orthogonality-constrained problems via geodesic line search.
General-relativistic GPS — clock corrections require integrating the metric's time dilation along satellite worldlines; errors accumulate without curvature corrections.

Historical Development

1827 — Gauss's Disquisitiones generales proves the Theorema Egregium: Gaussian curvature of a surface is intrinsic, depending only on the first fundamental form.
1854 — Riemann's habilitation lecture introduces $n$ -dimensional manifolds, the line element $d s^{2} = g_{ij} d x^{i} d x^{j}$ , and the curvature obstruction to flatness.
1869–1873 — Christoffel and Lipschitz introduce the symbols $Γ_{ij}^{k}$ and study covariant transformation laws.
1900 — Ricci-Curbastro and Levi-Civita publish the absolute differential calculus (tensor calculus), making the subject computational.
1915–1917 — Einstein's general relativity adopts pseudo-Riemannian geometry; Levi-Civita formalizes parallel transport, clarifying the connection.
1917–1923 — Élie Cartan develops the moving-frame method, exterior calculus, and the structure equations $d ω = - ω \land ω + Ω$ .
1926–1935 — Cartan classifies symmetric spaces; Hopf–Rinow (1931) settles completeness; Bonnet–Myers links Ricci curvature to diameter.
1950s–1960s — Berger classifies holonomy groups; comparison theorems (Rauch, Toponogov, Bishop–Gromov) mature; Nash's embedding theorem (1956) shows every Riemannian manifold sits isometrically in some $R^{N}$ .
1981–present — Gromov's metric-geometry program, Cheeger–Gromov convergence, Hamilton's Ricci flow (1982) and Perelman's 2003 resolution of the Poincaré and geometrization conjectures put curvature flows at the center.

Subtleties & Limits

Several deep questions remain partly open. The prescribed curvature problem — which functions arise as the scalar curvature of a metric on a given manifold (Yamabe-type and Kazdan–Warner constraints) — is fully solved only in special cases. The relationship between positive sectional curvature and topology is notoriously incomplete: very few manifolds are known to admit metrics of positive sectional curvature, and the gap between the known examples and obstructions (Hopf conjectures: does $S^{2} \times S^{2}$ admit positive curvature? is the Euler characteristic of an even-dimensional positively curved manifold positive?) is wide. Collapse and convergence of Riemannian manifolds with bounded curvature (Cheeger–Fukaya–Gromov theory) still hides structural mysteries in higher dimensions. Finally, the degenerate/singular limits of Ricci flow — characterizing all possible singularity models in dimensions $\geq 4$ — remain an active frontier, central to potential extensions of geometrization.