Chapter 1: Riemannian Geometry

Differential geometry of curved spaces

Goal of this page

Study the Levi-Civita connection and the curvature tensor in detail, and understand the central concepts of Riemannian geometry such as Jacobi fields and comparison theorems.

Riemannian geometry is the branch of differential geometry that equips each point of a curved space with an inner product (a Riemannian metric) and, on that basis, defines distance, angle, curvature, and geodesics (shortest paths) intrinsically within differential geometry. It generalizes Euclidean geometry to surfaces and manifolds, and forms the mathematical foundation of general relativity. Below we look in turn at its core ingredients: the connection, curvature, Jacobi fields, and comparison theorems.

1. The Levi-Civita Connection

Theorem: existence and uniqueness of the Levi-Civita connection

For a Riemannian manifold $(M, g)$ there exists a unique connection $\nabla$ satisfying:

Torsion-free: $\nabla_X Y - \nabla_Y X = [X, Y]$
Metric-compatible: $X g(Y, Z) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)$

Definition: covariant derivative

The covariant derivative of a vector field $Y$ in the direction $X$:

$$\nabla_X Y = \displaystyle\sum_k \left( X(Y^k) + \displaystyle\sum_{i,j} \Gamma^k_{ij} X^i Y^j \right) \dfrac{\partial}{\partial x^k}$$

Intuition: the connection is a “comparator” for vectors

On flat space, vectors at two separated points can be compared by subtracting their components directly. But on a curved space the coordinate basis vectors $\partial/\partial x^k$ themselves change direction from point to point. The covariant derivative corrects for this change of basis using the Christoffel symbols $\Gamma^k_{ij}$, extracting only “how much the vector field itself changed.” The condition $\nabla_X Y = 0$ means that “$Y$ is kept parallel in the direction $X$,” which is the definition of parallel transport.

Definition: Christoffel symbols

The Levi-Civita connection is uniquely determined by the metric $g_{ij}$:

$$\Gamma^k_{ij} = \frac{1}{2} \sum_l g^{kl}\left( \frac{\partial g_{jl}}{\partial x^i} + \frac{\partial g_{il}}{\partial x^j} - \frac{\partial g_{ij}}{\partial x^l} \right)$$

Here $g^{kl}$ is the inverse of the metric tensor $g_{ij}$. This expression follows uniquely from the two conditions of being torsion-free and metric-compatible.

Worked example: Christoffel symbols of the polar-coordinate plane

Writing the plane in polar coordinates $(r, \theta)$, the metric is $ds^2 = dr^2 + r^2\, d\theta^2$, i.e. $g_{rr}=1,\ g_{\theta\theta}=r^2,\ g_{r\theta}=0$, with inverse metric $g^{rr}=1,\ g^{\theta\theta}=1/r^2$. Only $\partial_r g_{\theta\theta} = 2r$ is nonzero, so

$$\Gamma^r_{\theta\theta} = -\tfrac{1}{2} g^{rr}\,\partial_r g_{\theta\theta} = -r,\qquad \Gamma^\theta_{r\theta} = \Gamma^\theta_{\theta r} = \tfrac{1}{2} g^{\theta\theta}\,\partial_r g_{\theta\theta} = \frac{1}{r}$$

All other components are $0$. Even though the plane is flat, the Christoffel symbols do not vanish — this reflects not curvature of the space but “distortion of the curvilinear coordinates.” Indeed, computing the curvature tensor of the next section gives all zeros, confirming that the plane is flat.

Once a connection is fixed, we can define parallel transport of a vector along a curve. Writing the transported tangent vector as $V$, the curve as $\gamma$, and its tangent (velocity vector) as $\gamma'$, the vector $V$ is carried while satisfying $\nabla_{\gamma'} V = 0$. Here “parallel” means that it does not rotate relative to the surface (an inhabitant of the surface always sees it pointing the same way); on a plane this coincides with ordinary parallel translation that carries the vector without changing its direction. When the surface is curved, however, the tangent plane itself tilts, so viewed from the surrounding 3-dimensional space $V$ appears to change direction little by little. This “apparent rotation” is precisely the manifestation of curvature (see the figure below).

A 3D figure of parallel transport along a curve γ on a curved surface: the tangent vector V is carried keeping a constant angle with the tangent γ′, changing direction according to the curvature of the surface. — Figure 1: Parallel transport along a curve $\gamma$ on a surface. The tangent vector $V$ is carried keeping a constant angle with the tangent $\gamma'$ ($\nabla_{\gamma'}V = 0$), changing direction according to the curvature of the surface.

2. Properties of the Curvature Tensor

Definition: Riemann curvature tensor

$$R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z$$

Intuition: curvature = path-dependence of parallel transport

Second covariant derivatives do not commute in general. That discrepancy $\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}$ is the curvature $R(X,Y)$. Geometrically it works as follows. In Figure 1 we carried $V$ along an open curve; here we parallel transport the same tangent vector $V$ once around a closed curve through a point $P$ and back to $P$. Calling the returned vector $V'$, on a curved space $V'$ is rotated from the original $V$ by an angle $\Delta\theta$ (holonomy), and that rotation angle $\Delta\theta$ equals the total curvature enclosed by the loop. If the curvature is zero the rotation is zero too — on flat space parallel transport is path-independent (see the figure below).

A 3D figure: transporting a vector parallel along a latitude circle (closed curve) on a sphere rotates V by Δθ once it returns to the start point. — Figure 2: Holonomy. Transporting parallel once around a closed curve on a surface rotates $V$ by $\displaystyle\Delta\theta = \iint_\Sigma K\,dA$.

Why the angle $\Delta\theta$ in the figure (the angle between $V$ and $V'$) equals curvature

Parallel transport around a closed curve is a rotation of the tangent plane, so the returned vector is $V' = R_{\Delta\theta}\,V$ with unchanged length ($|V'|=|V|$). Hence the angle formed by $V$ (start) and $V'$ (after one loop) is exactly the rotation angle:

$$\Delta\theta = \cos^{-1}\!\frac{\langle V',\,V\rangle}{|V'|\,|V|} = \iint_\Sigma K\, dA$$

The left equality is the geometric fact that “$V'$ is $V$ rotated by $\Delta\theta$”; the right equality is the local Gauss-Bonnet theorem (rotation angle = total curvature enclosed). However, $\cos^{-1}$ returns a value in $[0,\pi]$, so this identity holds directly only when $0 \le \Delta\theta \le \pi$. When the total curvature is large and $\Delta\theta$ exceeds $\pi$, or when the orientation (sign) of the rotation is also taken into account, $\iint_\Sigma K\,dA$ carries the complete information ($\cos^{-1}$ folds back to the principal value).

Theorem: symmetries of the curvature tensor

$R(X, Y) = -R(Y, X)$
$g(R(X, Y)Z, W) = -g(R(X, Y)W, Z)$
$g(R(X, Y)Z, W) = g(R(Z, W)X, Y)$
Bianchi identity: $R(X, Y)Z + R(Y, Z)X + R(Z, X)Y = 0$

Definition: sectional curvature

The sectional curvature of a 2-dimensional subspace $\sigma = \text{span}\{X, Y\}$:

$$K(\sigma) = \dfrac{g(R(X, Y)Y, X)}{g(X, X)g(Y, Y) - g(X, Y)^2}$$

Intuition: sectional curvature is the bending of a cross-section

Choosing a 2-dimensional plane $\sigma$ inside the tangent space, $K(\sigma)$ is the (Gaussian) curvature of the surface swept out by the geodesics in those directions. The sign determines the geometry: $K>0$ is spherical (geodesics converge; the angle sum of a geodesic triangle exceeds $\pi$), $K=0$ is flat, and $K<0$ is hyperbolic (geodesics diverge; the angle sum is less than $\pi$).

A geodesic triangle on a sphere; the edges bulge outward and the angle sum exceeds π. — Figure 3: The sign of the sectional curvature appears in the angle sum of a geodesic triangle. For $K>0$ the edges bulge outward, for $K=0$ they are straight, and for $K<0$ they cave inward.

A triangle on the plane; the edges are straight and the angle sum is π. — Figure 3: The sign of the sectional curvature appears in the angle sum of a geodesic triangle. For $K>0$ the edges bulge outward, for $K=0$ they are straight, and for $K<0$ they cave inward.

Worked example: constant-curvature spaces

The sphere $S^2_r$ of radius $r$ has constant sectional curvature $K = 1/r^2$ in every 2-dimensional direction. The larger the radius, the smaller the curvature, approaching the plane ($K\to 0$) as $r\to\infty$. On the other hand, the hyperbolic plane $\mathbb{H}^2$ is a constant-curvature space with $K=-1$. These three — the sphere ($K>0$), the Euclidean plane ($K=0$), and the hyperbolic plane ($K<0$) — are the representatives of constant-curvature spaces, corresponding to the three triangles in Figure 3.

3. Jacobi Fields

Definition: Jacobi field

A vector field $J$ arising from a variation of geodesics that satisfies the Jacobi equation:

$$\nabla_{\gamma'}^2 J + R(J, \gamma')\gamma' = 0$$

is called a Jacobi field.

Intuition: a Jacobi field is the “spread” of geodesics

Launching geodesics from a single point with slightly varying directions, they separate and then re-gather. The Jacobi field $J$ measures the gap between neighboring geodesics, and its growth is governed by the Jacobi equation. In two dimensions the equation takes the form $J'' + K J = 0$, where the sectional curvature $K$ plays the role of a spring constant. If $K>0$, $J$ oscillates and returns to $0$ (a conjugate point — the re-focusing of geodesics); if $K<0$, it diverges exponentially.

A semi-transparent sphere viewed head-on. From the left and right ends p and q (antipodal points), the reference geodesic γ and a neighbor geodesic spread on the front face and re-gather at the conjugate point q. The Jacobi field J is their gap, zero at both ends and maximal in the middle. — Figure 4: Jacobi field and conjugate points. Placing $p, q$ (antipodal points) at the left and right ends of a semi-transparent sphere makes both clearly visible. The reference geodesic $\gamma$ and a neighbor geodesic leaving $p$ open up on the front face and re-gather at the conjugate point $q$. $J$ is their gap (zero at both ends).

Theorem: conjugate points

A value $t_0 > 0$ for which a nonzero Jacobi field $J \neq 0$ has $J(0) = 0$ and $J(t_0) = 0$ is called a conjugate point.

Beyond a conjugate point the geodesic is no longer shortest.

Worked example: conjugate points on the sphere

On the unit sphere $S^2$ ($K=1$), the Jacobi equation along a great circle is $J'' + J = 0$. The solution with $J(0)=0$ is $J(t) = \sin t$, which returns to $0$ at $t = \pi$. That is, geodesics (meridians) leaving the north pole meet again at the antipodal point (the point diametrically opposite through the center — the south pole for the north pole) at distance $\pi$, and that is the conjugate point. For a sphere of radius $r$ the distance to the conjugate point is $\pi r$ (the antipode). The way geodesics open up and re-gather at the conjugate point is exactly the lens shape of Figure 4 (Figure 4 uses a flattened surface of revolution to make $p, q$ easy to see).

4. Comparison Theorems

Theorem: Bonnet-Myers theorem

If the Ricci curvature satisfies $\text{Ric} \geq (n-1)\kappa > 0$, then $M$ is compact and

$$\text{diam}(M) \leq \dfrac{\pi}{\sqrt{\kappa}}$$

Intuition: positive curvature confines space

If the Ricci curvature has a positive lower bound, then every geodesic, in whatever direction, reaches a conjugate point within a finite distance and is no longer shortest beyond it. Hence the manifold cannot extend infinitely: its diameter is bounded by $\pi/\sqrt{\kappa}$ and it is compact. Picture curvature acting as a “restoring force that pulls geodesics back”; the oscillation (re-focusing) of the Jacobi equation $J''+KJ=0$ from the previous section is its essence.

Worked example: equality on the sphere

The sphere $S^2$ of radius $r$ has $\mathrm{Ric} = (n-1)/r^2$ ($n=2$), i.e. $\kappa = 1/r^2$. The Bonnet-Myers bound is $\mathrm{diam} \le \pi/\sqrt{\kappa} = \pi r$. The actual diameter of the sphere (half the length of a great circle) is exactly $\pi r$, so equality holds. The sphere is the typical equality case of the comparison theorem.

Theorem: Gauss-Bonnet theorem

For a compact 2-dimensional Riemannian manifold $M$:

$$\displaystyle\int_M K\, dA = 2\pi \chi(M)$$

Intuition: total curvature depends only on the number of holes

The Gauss-Bonnet theorem asserts that integrating the local bending $K$ over the whole surface gives a total that depends only on the topology of the surface (the Euler characteristic $\chi$). However bumpy the deformation, the total curvature is invariant. The sphere has $\chi=2$ with $\int K\,dA = 4\pi$; the torus (doughnut) has $\chi=0$ with $\int K\,dA = 0$ — the positive curvature on the outside and the negative curvature on the inside (the hole side) cancel exactly.

A sphere with uniformly positive curvature. — Figure 5: The total curvature $\int_M K\,dA = 2\pi\chi(M)$ is determined solely by topology. In the figure, **warm (orange) marks positive curvature $K>0$** and **cool (blue) marks negative curvature $K \lt 0$**. The sphere ($\chi=2$) is $K>0$ everywhere, giving $4\pi$. On the torus ($\chi=0$) the outer bulge ($K>0$, orange) and the inner side facing the hole ($K \lt 0$, blue) cancel exactly to $0$.

A torus (doughnut) with positive curvature on the outside and negative curvature on the inside. — Figure 5: The total curvature $\int_M K\,dA = 2\pi\chi(M)$ is determined solely by topology. In the figure, **warm (orange) marks positive curvature $K>0$** and **cool (blue) marks negative curvature $K \lt 0$**. The sphere ($\chi=2$) is $K>0$ everywhere, giving $4\pi$. On the torus ($\chi=0$) the outer bulge ($K>0$, orange) and the inner side facing the hole ($K \lt 0$, blue) cancel exactly to $0$.

Worked example: angular excess of a geodesic triangle

Applying the Gauss-Bonnet theorem to a geodesic triangle, the difference between the angle sum and $\pi$ (the angular excess) equals the total curvature inside:

$$(\alpha+\beta+\gamma) - \pi = \iint_{\triangle} K\, dA$$

Consider on the unit sphere a triangle whose three sides all meet at right angles (an octant of the sphere): then $\alpha=\beta=\gamma=\pi/2$, so the angular excess is $3\cdot\tfrac{\pi}{2}-\pi = \tfrac{\pi}{2}$. Its area is $1/8$ of the whole sphere $4\pi$, namely $\tfrac{\pi}{2}$, and since $K=1$ the right-hand side is also $\tfrac{\pi}{2}$, so the two sides agree. This quantifies the $K>0$ triangle of Figure 3 (angle sum $>\pi$).

Summary

Key points of this chapter

Levi-Civita connection: torsion-free + metric-compatible
Curvature tensor: path-dependence of parallel transport
Jacobi field: variation of geodesics, conjugate points
Comparison theorems: deriving global properties from the sign of curvature

This article is a survey overview of the branches of advanced geometry. To study the connection, curvature, Jacobi fields, and comparison theorems covered here chapter by chapter and in more depth — from the Riemannian metric through geodesics, the exponential map, and the curvature tensor — continue to the differential geometry course.

Frequently asked questions

What is Riemannian geometry?

Riemannian geometry is the branch of differential geometry that studies manifolds (Riemannian manifolds) equipped with an inner product (a Riemannian metric) at each point. It lets one define distance, angle, curvature, and geodesics (shortest paths) intrinsically, and forms the mathematical foundation of general relativity.

What is the Levi-Civita connection?

It is the standard (symmetric and metric-compatible) covariant derivative on a Riemannian manifold. It defines parallel transport and characterizes geodesics as curves of zero acceleration. It is expressed in local coordinates by the Christoffel symbols $\Gamma^k_{ij}$ and is used to compute the Riemann curvature tensor.

What is the relationship between the Riemann curvature tensor and the sectional curvature?

The Riemann curvature tensor $R^l{}_{kij}$ is a $(1,3)$-type tensor describing the bending of a Riemannian manifold. The sectional curvature $K(\sigma)$ is the cross-sectional curvature of a 2-dimensional subspace $\sigma$ of the tangent space, given by a particular contraction of the tensor. Constant-curvature spaces (sphere, hyperbolic space, flat space) are the special case where the sectional curvature is constant.