Riemannian Geometry
Riemannian geometry equips smooth manifolds with a metric tensor—an inner product on each tangent space that varies smoothly from point to point. This structure enables the measurement of lengths, angles, areas, and curvature, and provides the mathematical foundation for Einstein's general theory of relativity.
Historical Context
Bernhard Riemann introduced the concept of a metric tensor in his legendary 1854 Habilitationsschrift, delivered before Gauss at Göttingen. Riemann generalized Gauss's intrinsic geometry of surfaces to arbitrary dimensions, proposing that the geometry of a space should be determined by a quadratic form $ds^2 = g_{ij}\,dx^i\,dx^j$ on infinitesimal displacements. This was sixty years before Einstein used exactly this framework to formulate general relativity in 1915.
The Levi-Civita connection, developed by Tullio Levi-Civita in 1917, provided the canonical notion of parallel transport compatible with the Riemannian metric. Elwin Bruno Christoffel had earlier (1869) introduced the symbols that bear his name for expressing covariant differentiation in coordinates. Together, these tools form the computational backbone of modern differential geometry and general relativity.
The theory reached maturity through the work of Elie Cartan, who reformulated connections using moving frames, and Marcel Berger, who classified Riemannian holonomy groups.
Derivation 1: The Riemannian Metric Tensor
A Riemannian metric on a smooth manifold $M$ is a smooth assignment of an inner product $g_p$ on each tangent space $T_pM$. In local coordinates$\{x^i\}$, the metric is specified by a symmetric, positive-definite matrix:
$g = g_{ij}(x)\,dx^i \otimes dx^j$
The line element gives the infinitesimal distance:
$ds^2 = g_{ij}\,dx^i\,dx^j$
Transformation Law
Under a coordinate change $x^i \to \bar{x}^a$, the metric components transform as a (0,2)-tensor:
$\bar{g}_{ab} = \frac{\partial x^i}{\partial \bar{x}^a}\frac{\partial x^j}{\partial \bar{x}^b}\,g_{ij}$
Example: The 2-Sphere
In spherical coordinates $(\theta, \phi)$ on the unit sphere $S^2 \subset \mathbb{R}^3$:
$ds^2 = d\theta^2 + \sin^2\theta\,d\phi^2, \quad g = \begin{pmatrix} 1 & 0 \\ 0 & \sin^2\theta \end{pmatrix}$
General relativity: In GR, spacetime carries a pseudo-Riemannian (Lorentzian) metric of signature $(-,+,+,+)$. The Schwarzschild metric for a black hole of mass $M$ is $ds^2 = -(1-\frac{2GM}{rc^2})c^2dt^2 + (1-\frac{2GM}{rc^2})^{-1}dr^2 + r^2 d\Omega^2$.
Derivation 2: Christoffel Symbols
The Christoffel symbols of the second kind encode how basis vectors change from point to point. They are derived from the metric by requiring two conditions: metric compatibility and torsion-freedom.
The Koszul Formula
From the conditions $\nabla g = 0$ (metric compatible) and $\nabla_X Y - \nabla_Y X = [X,Y]$ (torsion-free), one derives the Koszul formula:
$2g(\nabla_X Y, Z) = X(g(Y,Z)) + Y(g(X,Z)) - Z(g(X,Y)) + g([X,Y],Z) - g([X,Z],Y) - g([Y,Z],X)$
Applying this to coordinate vector fields $X = \partial_i$, $Y = \partial_j$, $Z = \partial_k$(where $[\partial_i, \partial_j] = 0$):
$\boxed{\Gamma^k_{ij} = \frac{1}{2}g^{kl}\left(\frac{\partial g_{il}}{\partial x^j} + \frac{\partial g_{jl}}{\partial x^i} - \frac{\partial g_{ij}}{\partial x^l}\right)}$
Computation for $S^2$
With $g_{11} = 1$, $g_{22} = \sin^2\theta$, $g_{12} = 0$, the only non-vanishing partial derivative is $\partial_\theta g_{22} = 2\sin\theta\cos\theta$. This yields:
$\Gamma^\theta_{\phi\phi} = -\frac{1}{2}g^{\theta\theta}\partial_\theta g_{\phi\phi} = -\sin\theta\cos\theta$
$\Gamma^\phi_{\theta\phi} = \Gamma^\phi_{\phi\theta} = \frac{1}{2}g^{\phi\phi}\partial_\theta g_{\phi\phi} = \frac{\cos\theta}{\sin\theta} = \cot\theta$
Counting Christoffel symbols: In $n$ dimensions, there are $\frac{n^2(n+1)}{2}$ independent Christoffel symbols (symmetric in lower indices). In 4D spacetime, this gives 40 independent components.
Derivation 3: The Levi-Civita Connection
The fundamental theorem of Riemannian geometry states that on every Riemannian manifold $(M, g)$, there exists a unique affine connection $\nabla$ that is:
$\nabla_X g = 0 \quad \text{(metric compatible: inner products are preserved)}$
$T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y] = 0 \quad \text{(torsion-free)}$
Proof of Uniqueness
Suppose $\nabla$ and $\bar{\nabla}$ are both metric-compatible and torsion-free. Their difference $D(X,Y) = \nabla_X Y - \bar{\nabla}_X Y$ is a tensor (the non-tensorial parts cancel). From metric compatibility:
$g(D(X,Y), Z) + g(Y, D(X,Z)) = 0$
From the torsion-free condition, $D(X,Y) = D(Y,X)$. Cycling the metric compatibility equation over$(X,Y,Z)$ and using symmetry, one obtains $g(D(X,Y),Z) = 0$ for all $Z$. Since $g$ is non-degenerate, $D = 0$, proving uniqueness.
Covariant Derivative
The covariant derivative of a vector field $V = V^i \partial_i$ along a direction $\partial_j$ is:
$\nabla_j V^i = \partial_j V^i + \Gamma^i_{jk} V^k$
This extends to arbitrary tensors. For a (0,2)-tensor such as the metric:$\nabla_k g_{ij} = \partial_k g_{ij} - \Gamma^l_{ki} g_{lj} - \Gamma^l_{kj} g_{il} = 0$, which is precisely the metric compatibility condition.
Derivation 4: Parallel Transport and Holonomy
A vector field $V$ along a curve $\gamma(t)$ is parallel transported if its covariant derivative along the curve vanishes:
$\frac{DV^i}{dt} = \frac{dV^i}{dt} + \Gamma^i_{jk}\frac{dx^j}{dt}V^k = 0$
This is a system of first-order ODEs. Given an initial vector $V(0) \in T_{\gamma(0)}M$, there exists a unique parallel transport along $\gamma$. The resulting map:
$P_\gamma: T_{\gamma(0)}M \to T_{\gamma(1)}M$
is a linear isomorphism. For the Levi-Civita connection, it is also an isometry (preserves inner products).
Holonomy on $S^2$
When a vector is parallel-transported around a closed loop, it generally returns rotated. For a loop enclosing solid angle $\Omega$ on $S^2$, the rotation angle equals $\Omega$. For a latitude circle at colatitude $\theta$:
$\boxed{\Delta\alpha = \oint K\,dA = 2\pi(1 - \cos\theta)}$
Physical realization: The Foucault pendulum demonstrates holonomy on Earth. At latitude $\lambda$, the pendulum's plane rotates by $2\pi\sin\lambda$ per sidereal day—precisely the holonomy of parallel transport around the latitude circle on the sphere.
Derivation 5: Index Raising, Lowering, and Musical Isomorphisms
The metric provides a canonical isomorphism between vectors and covectors. The flat ($\flat$) and sharp ($\sharp$) operations (named for musical notation) allow index manipulation:
$\flat: TM \to T^*M, \quad V \mapsto V^\flat, \quad V^\flat_i = g_{ij}V^j \quad \text{(lowering)}$
$\sharp: T^*M \to TM, \quad \omega \mapsto \omega^\sharp, \quad \omega^{\sharp i} = g^{ij}\omega_j \quad \text{(raising)}$
Here $g^{ij}$ is the inverse metric satisfying $g^{ik}g_{kj} = \delta^i_j$. These operations extend to arbitrary tensors, enabling free movement between covariant and contravariant descriptions.
The Gradient as a Sharp
The gradient of a function $f$ is the sharp of its differential:
$(\text{grad}\, f)^i = g^{ij}\partial_j f = (df)^{\sharp i}$
In flat Cartesian coordinates, $g^{ij} = \delta^{ij}$ and this reduces to the familiar$\nabla f = (\partial_x f, \partial_y f, \partial_z f)$. But in curvilinear coordinates (spherical, cylindrical), the metric factors produce the well-known correction terms.
In GR: Raising and lowering indices with the spacetime metric$g_{\mu\nu}$ is fundamental. The Einstein equation $G_{\mu\nu} = 8\pi G\,T_{\mu\nu}$ involves the Ricci tensor and scalar obtained by contracting the Riemann tensor using the inverse metric.
Interactive Simulation
This simulation computes the metric tensor, Christoffel symbols, and parallel transport on the 2-sphere. It demonstrates how vectors rotate during parallel transport around latitude circles and verifies the holonomy formula.
Riemannian Geometry: Metric, Christoffel Symbols & Parallel Transport
PythonClick Run to execute the Python code
Code will be executed with Python 3 on the server
Summary
Metric Tensor
The symmetric positive-definite tensor $g_{ij}$ defines lengths, angles, and volumes on a Riemannian manifold. It transforms as a (0,2)-tensor under coordinate changes.
Levi-Civita Connection
The unique torsion-free, metric-compatible connection. Its coefficients are the Christoffel symbols, determined entirely by the metric and its first derivatives.
Parallel Transport
Parallel transport along curves preserves inner products (for Levi-Civita). On curved spaces, transport around closed loops produces a rotation (holonomy) that encodes curvature.
Musical Isomorphisms
The metric provides canonical identifications between vectors and covectors via index raising and lowering, unifying contravariant and covariant descriptions.
Practice Problems
Problem 1: Christoffel Symbols for the 2-SphereThe metric on $S^2$ of radius $R$ in coordinates $(\theta, \phi)$ is $ds^2 = R^2\,d\theta^2 + R^2\sin^2\!\theta\,d\phi^2$. Compute all non-vanishing Christoffel symbols.
Solution:
1. The metric components are $g_{\theta\theta} = R^2$, $g_{\phi\phi} = R^2\sin^2\!\theta$, with $g_{\theta\phi} = 0$. The inverse metric: $g^{\theta\theta} = 1/R^2$, $g^{\phi\phi} = 1/(R^2\sin^2\!\theta)$.
2. Using $\Gamma^{\lambda}_{\mu\nu} = \frac{1}{2}g^{\lambda\rho}(\partial_\mu g_{\nu\rho} + \partial_\nu g_{\mu\rho} - \partial_\rho g_{\mu\nu})$. The only non-zero derivative is $\partial_\theta g_{\phi\phi} = 2R^2\sin\theta\cos\theta$.
3. Compute $\Gamma^\theta_{\phi\phi}$:
4. Compute $\Gamma^\phi_{\theta\phi} = \Gamma^\phi_{\phi\theta}$:
5. All other Christoffel symbols vanish. Summary:
Note that $R$ cancels entirely — the Christoffel symbols are independent of the sphere's radius, reflecting the fact that they encode the intrinsic shape (curvature ratios), not the overall scale.
Problem 2: Parallel Transport Around a Loop on $S^2$A vector initially pointing east is parallel transported along a closed triangular loop on the unit sphere: from the north pole along the 0° meridian to the equator, along the equator to longitude 90°, then back to the north pole. Find the angle by which the vector has rotated.
Solution:
1. The holonomy (rotation angle) from parallel transport around a closed loop on a surface of constant curvature equals the solid angle (area) enclosed:
2. For the unit sphere, the Gaussian curvature is $K = 1/R^2 = 1$ everywhere.
3. The triangular loop encloses one-eighth of the sphere (bounded by two meridians 90° apart and the equator). The area of a spherical lune of angle $\Delta\phi$ between equator and pole is $\Delta\phi$ (in steradians on the unit sphere):
4. The holonomy angle equals the enclosed area:
5. After traversing the loop, the vector has rotated 90° relative to its initial orientation. This is a direct manifestation of the curvature of $S^2$: parallel transport on a flat surface around any closed loop returns a vector unchanged. The deficit angle equals the integrated curvature, which is the content of the Ambrose-Singer theorem for Riemannian holonomy.
Problem 3: Geodesic Equation in Polar CoordinatesWrite the geodesic equations for the flat plane in polar coordinates $(r, \phi)$ with metric $ds^2 = dr^2 + r^2\,d\phi^2$. Verify that straight lines satisfy these equations.
Solution:
1. First compute the Christoffel symbols. With $g_{rr} = 1$, $g_{\phi\phi} = r^2$, the non-vanishing symbols are:
2. The geodesic equations $\ddot{x}^\mu + \Gamma^\mu_{\alpha\beta}\dot{x}^\alpha\dot{x}^\beta = 0$ are:
3. The second equation can be rewritten as $\frac{1}{r^2}\frac{d}{d\lambda}(r^2\dot{\phi}) = 0$, giving the conserved angular momentum:
4. A straight line $y = b$ (constant) in Cartesian corresponds to $r\sin\phi = b$ in polar. Differentiating: $\dot{r}\sin\phi + r\dot{\phi}\cos\phi = 0$. One can verify $r^2\dot{\phi} = b\,v$ (constant) along this path, confirming the angular momentum conservation.
5. Substituting back into the radial equation and using $\dot{\phi} = \ell/r^2$:
This is the centripetal acceleration needed to maintain a straight-line path in polar coordinates — purely a coordinate artifact on flat space, illustrating how Christoffel symbols encode coordinate effects rather than true curvature.
Problem 4: Ricci Scalar for the 2-SphereUsing the Christoffel symbols from Problem 1, compute the Riemann tensor component $R^\theta{}_{\phi\theta\phi}$, the Ricci tensor, and the Ricci scalar for the 2-sphere of radius $R$.
Solution:
1. The Riemann tensor is $R^\lambda{}_{\sigma\mu\nu} = \partial_\mu\Gamma^\lambda_{\nu\sigma} - \partial_\nu\Gamma^\lambda_{\mu\sigma} + \Gamma^\lambda_{\mu\rho}\Gamma^\rho_{\nu\sigma} - \Gamma^\lambda_{\nu\rho}\Gamma^\rho_{\mu\sigma}$. Compute $R^\theta{}_{\phi\theta\phi}$:
2. Evaluating each term with $\Gamma^\theta_{\phi\phi} = -\sin\theta\cos\theta$ and $\Gamma^\phi_{\theta\phi} = \cot\theta$:
3. The Ricci tensor: $R_{\phi\phi} = R^\theta{}_{\phi\theta\phi} = \sin^2\!\theta$ and $R_{\theta\theta} = g^{\phi\phi}R_{\theta\phi\theta\phi} = \frac{1}{R^2\sin^2\theta}\cdot R^2\sin^2\theta = 1$.
4. The Ricci scalar is:
5. Therefore:
This is constant over the sphere, as expected for a maximally symmetric space. The Gaussian curvature is $K = \mathcal{R}/2 = 1/R^2$. For the unit sphere, $\mathcal{R} = 2$, and the Gauss-Bonnet theorem gives $\int \mathcal{R}\,dA = 4\pi\chi(S^2) = 8\pi$.
Problem 5: Geodesic Deviation on the SphereTwo geodesics (great circles) on the unit sphere start at the equator separated by a small distance $\xi_0$, both heading north. Using the geodesic deviation equation, find their separation as a function of latitude $\theta$.
Solution:
1. The geodesic deviation (Jacobi) equation describes how nearby geodesics converge or diverge:
2. For meridians (constant $\phi$ lines) on the unit sphere, the tangent vector is $\dot{x}^\alpha = (1, 0)$ in $(\theta, \phi)$ coordinates, and the separation vector is along $\phi$: $\xi^\mu = (0, \xi)$.
3. The relevant curvature component is $R^\phi{}_{\theta\phi\theta} = -\sin^2\theta/(R^2\sin^2\theta) \cdot R^2 = -1$ (for unit sphere). The deviation equation reduces to:
4. This is SHM with solution $\xi(\theta) = A\cos\theta + B\sin\theta$. With initial conditions $\xi(\pi/2) = \xi_0$ (at equator) and $d\xi/d\theta = 0$ (parallel start):
5. The separation as a function of colatitude is:
At $\theta = 0$ (north pole), $\xi = 0$: the meridians converge to a point, which is the physical separation of great circles meeting at the pole. This focusing of geodesics is a hallmark of positive curvature and is the geometric content of the Rauch comparison theorem.