Manifolds 34 | Examples for Riemannian Manifolds [dark version]

TL;DR

A Riemannian metric is a smoothly varying inner product on each tangent space of a manifold.

Briefing Cornell Notes

Briefing

Riemannian manifolds get their geometry from a smoothly varying inner product on each tangent space, and the cleanest way to see how that works is through concrete submanifolds sitting inside Euclidean space. For any submanifold M ⊂ R^N, the surrounding Euclidean (dot-product) geometry induces a “standard” Riemannian metric on M: tangent vectors inherit their inner products from R^N. In coordinates, if a parameterization f describes M, then the metric components are computed by taking inner products of partial derivatives of f. This produces a matrix G(p) whose entries are G_ij(p) = ⟨∂f/∂x_i, ∂f/∂x_j⟩, evaluated at the point corresponding to p. Once G is known, lengths and areas follow from it—most notably through the square root of det(G), which acts as the scaling factor inside integrals.

The transcript then walks through two illustrative examples. First comes a one-dimensional manifold (a curve) in R^N. With only one parameter t, the tangent space is one-dimensional, so the metric reduces to a 1×1 matrix. The metric value is simply the squared Euclidean norm of the derivative of the parameterization: G_11 = ||f′(t)||^2. The length of the curve from t = A to t = B becomes an integral of the square root of that metric contribution, yielding the familiar formula involving ||f′(t)||.

Next, the discussion turns to the two-dimensional sphere S^2 embedded in R^3. Using spherical coordinates with angles θ and φ, the parameterization gives two tangent directions corresponding to partial derivatives with respect to θ and φ. Taking inner products of these tangent vectors produces a 2×2 metric matrix G. After simplifying trigonometric terms (notably sin^2 and cos^2 identities), the metric becomes diagonal with the only nontrivial entry equal to sin^2(θ) in the appropriate component. The determinant of this metric matrix then leads to √det(G) = |sin(θ)|.

That √det(G) factor is the key bridge to integration on manifolds. For two-dimensional surfaces like S^2, it supplies the correct area scaling in surface integrals. The transcript also foreshadows the next step: combining √det(G) with an appropriate differential form (a volume form) to compute areas and other integrals on S^2. The broader takeaway is that even when manifolds are abstract, the same metric-driven scaling principle lets them carry meaningful notions of length, area, and volume.

Cornell Notes

A Riemannian metric assigns to each point of a manifold an inner product on its tangent space, varying smoothly from point to point. For a submanifold M ⊂ R^N, the Euclidean inner product in the ambient space induces a natural “standard” metric on M: metric components come from inner products of partial derivatives of a parameterization f. In one dimension, the metric collapses to G_11 = ||f′(t)||^2, and curve length becomes an integral involving ||f′(t)||. For the sphere S^2 ⊂ R^3 with spherical coordinates (θ, φ), the induced metric is diagonal and yields √det(G) = |sin(θ)|. That determinant factor is the scaling needed for area and other surface integrals, motivating volume forms on manifolds.

How does a submanifold M ⊂ R^N automatically inherit a Riemannian metric?

Because tangent vectors to M can be viewed as vectors in the ambient Euclidean space. If f parameterizes M, then tangent vectors at a point correspond to partial derivatives of f. The metric components are computed by taking Euclidean inner products of these partial derivatives: G_ij(p) = ⟨∂f/∂x_i, ∂f/∂x_j⟩ at the point corresponding to p. This makes angles and lengths on M match the ambient Euclidean geometry.

Why does the metric for a 1D manifold reduce to a single quantity?

A one-dimensional manifold has a one-dimensional tangent space, so there is only one basis tangent vector direction. With parameter t, the metric becomes a 1×1 matrix: G_11 = ⟨f′(t), f′(t)⟩ = ||f′(t)||^2. Consequently, the length of the curve from A to B uses the norm of f′(t) inside an integral.

What is the induced metric on S^2 ⊂ R^3 in spherical coordinates?

Using spherical coordinates (θ, φ), the parameterization produces two tangent vectors: one from ∂/∂θ and one from ∂/∂φ. Taking inner products of these tangent vectors yields a 2×2 metric matrix G. After trigonometric simplification, the metric is diagonal with the only nontrivial component equal to sin^2(θ) (with the other off-diagonal and diagonal terms becoming 0 or 1 as computed).

How does √det(G) connect to measuring length and area on manifolds?

In the integration formulas, √det(G) acts as the scaling factor coming from the metric. For curves, the metric contribution reduces to ||f′(t)|| (equivalently derived from the 1×1 determinant). For surfaces like S^2, √det(G) becomes |sin(θ)|, which supplies the correct area element scaling when integrating over θ and φ.

Why does the discussion foreshadow volume forms after computing √det(G)?

√det(G) provides the metric-dependent scaling, but integration on manifolds also needs the right differential-form structure (a volume form). The transcript indicates that √det(G) will be combined with a two-form to produce the area element on S^2, enabling integrals over the surface using the manifold’s intrinsic geometry.

Review Questions

Given a parameterization f of a submanifold M ⊂ R^N, how do you compute the metric matrix entries G_ij?
For a curve parameterized by t, what does the induced metric reduce to, and how does it enter the length integral?
For S^2 with spherical coordinates (θ, φ), what is √det(G) and why is it important for surface integration?

Key Points

1
A Riemannian metric is a smoothly varying inner product on each tangent space of a manifold.
2
Any submanifold M ⊂ R^N inherits a standard Riemannian metric from the ambient Euclidean inner product.
3
With a parameterization f, metric components are computed as inner products of partial derivatives of f.
4
For a one-dimensional manifold (a curve), the metric becomes G_11 = ||f′(t)||^2 and curve length follows from integrating ||f′(t)||.
5
For S^2 ⊂ R^3 in spherical coordinates, the induced metric is diagonal and has a nontrivial sin^2(θ) component.
6
The quantity √det(G) supplies the metric-dependent scaling factor needed for length/area integrals on manifolds.
7
Volume forms will use √det(G) together with differential forms to produce the correct intrinsic area element on S^2.

Highlights

For submanifolds M ⊂ R^N, the Euclidean dot product induces the metric on M by taking inner products of tangent vectors coming from parameter derivatives.

A curve’s induced metric collapses to a single value: G_11 = ||f′(t)||^2, making length an integral of ||f′(t)||.

On S^2 with spherical coordinates, the induced metric yields √det(G) = |sin(θ)|, the essential factor for surface area integration.

Computing √det(G) is the gateway to building volume forms that enable integration on manifolds beyond Euclidean space.

Topics

Riemannian Metrics
Submanifolds in Euclidean Space
Induced Metric
Spherical Coordinates
Volume Forms