Questions: Riemannian Metrics

Question 1

The standard metric on ℝ² in polar coordinates (r, θ) is ds² = dr² + r²dθ². The coefficient r² in front of dθ² means that...

• A. The θ-coordinate curves have length r²
• B. A small displacement dθ at radius r corresponds to an actual distance of r·dθ, so distances in the θ-direction scale with r
• C. The metric is not flat because the coefficients are not constant
• D. The Gaussian curvature is 1/r²

Answer: (B)

The metric component gθθ = r² means that the length element in the θ-direction is |ds| = r|dθ|. A circle at radius r has circumference ∫₀²π r dθ = 2πr, as expected. The non-constant coefficient does NOT mean the metric is curved — flat ℝ² has curvature zero regardless of the coordinate system used. The appearance of r² is a coordinate artifact. This illustrates a key principle: the metric components gij depend on the coordinate system, but geometric quantities (curvature, geodesics, distances) do not.

Question 2

A Riemannian metric g defines a natural isomorphism between the tangent and cotangent spaces. This isomorphism, called the 'musical isomorphism,' maps a vector field X to the 1-form...

• A. X♭ = g(X, ·), defined by X♭(Y) = g(X, Y) for all Y
• B. X♭ = dX, the exterior derivative of X
• C. X♭ = X/|X|, the unit vector in the direction of X
• D. X♭ = ∇X, the covariant derivative of X

Answer: (A)

The 'flat' operator ♭ sends a vector X to the 1-form X♭ defined by X♭(Y) = g(X,Y). In coordinates, if X = Xⁱ∂/∂xⁱ, then X♭ = gijXⁱ dxʲ — the metric 'lowers the index.' The inverse operation ♯ ('sharp') raises indices: ω♯ is the vector field satisfying g(ω♯, Y) = ω(Y). For example, the gradient ∇f = (df)♯ is obtained by applying sharp to the differential df. Without a metric, there is no natural way to convert vectors to covectors.

Question 3

Why does the choice of Riemannian metric matter so much, given that every smooth manifold admits one?

Model answer: While existence is guaranteed, different metrics produce completely different geometric structures: different notions of distance, different geodesics, different curvature, different volume. A sphere with the round metric has positive curvature and finite diameter; with a flat metric (impossible globally, but illustrative locally) it would have zero curvature. The metric determines all of Riemannian geometry — connections, curvature, geodesics, the Laplacian — so choosing the 'right' metric for a given problem is the central question. In general relativity, the metric encodes the gravitational field.

The partition-of-unity construction proves existence but gives a 'generic' metric with no particular geometric significance. Interesting Riemannian geometry comes from metrics with special properties: constant curvature (spheres, hyperbolic space), Einstein metrics (Ricci curvature proportional to metric), Kahler metrics (compatible with complex structure). Finding metrics with prescribed curvature properties is one of the deepest problems in differential geometry.

Question 4

On a Riemannian manifold (M, g), the length of a smooth curve γ : [a,b] → M is defined as L(γ) = ∫_a^b √g(γ'(t), γ'(t)) dt.

• T. True
• F. False

Answer: True

The metric g provides an inner product on each tangent space, so g(γ'(t), γ'(t)) is the squared norm of the velocity vector. Its square root is the speed, and integrating speed over time gives arc length. This is the manifold generalization of the familiar formula L = ∫√(dx² + dy²) for curves in the plane. The distance between two points is the infimum of lengths of all curves connecting them, making (M, d) a metric space (in the topology sense) when M is connected.