Quiz: Lecture 15 — Riemannian Manifolds: Foundations

This quiz tests your understanding of smooth manifolds, tangent spaces, Riemannian metrics, exponential/logarithmic maps, curvature, and the SPD cone and sphere as Riemannian manifolds. There are 10 multiple-choice questions, each worth 1 mark. After submitting, your score is shown, and any questions you got wrong are highlighted — but the correct answer is not revealed. You can then redo only the questions you missed.

Q1. A topological manifold of dimension \(m\) is a topological space that is:

A. A compact metric space B. Any subset of \(\mathbb{R}^m\) C. Hausdorff, second countable, and locally homeomorphic to \(\mathbb{R}^m\) — every point has a neighborhood that looks like an open subset of \(\mathbb{R}^m\) D. Any vector space of dimension \(m\)

Q2. The tangent space \(T_p\mathcal{M}\) at a point \(p\) on a smooth manifold is:

A. The set of all points at distance \(\varepsilon\) from \(p\) B. A vector space of dimension \(\dim\mathcal{M}\) consisting of all tangent vectors (derivations or velocity vectors of curves) at \(p\) — it is the linearization of the manifold near \(p\) C. The set of all geodesics passing through \(p\) D. A curved subspace of the manifold

Q3. A Riemannian metric \(g\) on a manifold \(\mathcal{M}\) is:

A. The distance function \(d(p,q)\) between points B. A measure of the manifold’s volume C. A smooth assignment of an inner product \(g_p(\cdot, \cdot)\) to each tangent space \(T_p\mathcal{M}\), defining lengths, angles, and geodesics D. The same as a coordinate chart

Q4. The exponential map \(\exp_p : T_p\mathcal{M} \to \mathcal{M}\) and logarithmic map \(\log_p : \mathcal{M} \to T_p\mathcal{M}\) satisfy:

A. \(\exp_p(\log_p(q)) = q\) away from the cut locus of \(p\) — they are local inverses, and the relationship fails at the cut locus where geodesics from \(p\) cease to be minimizing B. \(\exp_p(\log_p(q)) = q\) for all \(q \in \mathcal{M}\) — they are global inverses on any manifold C. \(\exp_p\) and \(\log_p\) are unrelated functions D. They are always the same as in Euclidean space

Q5. The differential (pushforward) \(dF_p: T_p\mathcal{M} \to T_{F(p)}\mathcal{N}\) of a smooth map \(F: \mathcal{M} \to \mathcal{N}\) is:

A. The same map as \(F\), restricted to tangent vectors B. A linear map that sends a tangent vector \(v \in T_p\mathcal{M}\) to the velocity of the image curve \(F \circ \gamma\) at \(F(p)\), generalizing the Jacobian from multivariable calculus C. A measure of the distance between \(p\) and \(F(p)\) D. Always invertible for any smooth map \(F\)

Q6. Sectional curvature on a Riemannian manifold measures:

A. The dimension of the manifold B. How geodesics spreading from a point converge or diverge — it is the Gaussian curvature of the 2D surface spanned by two tangent directions C. The number of holes in the manifold D. The total volume of the manifold

Q7. A Hadamard manifold is a complete, simply connected Riemannian manifold with:

A. Zero curvature everywhere — it must be flat B. Positive sectional curvature everywhere C. Constant positive curvature D. Nonpositive sectional curvature everywhere — this guarantees unique geodesics between any two points, convex distance functions, and unique Fréchet means

Q8. The sphere \(\mathbb{S}^2\) has constant positive curvature. As a result:

A. It has the same geometry as Euclidean space B. Geodesics can focus — great circles from a point reconverge at the antipodal point, meaning the cut locus is nontrivial and Fréchet means are not globally unique C. It is a Hadamard manifold D. Geodesics are unique between any two points

Q9. The SPD cone \(\mathcal{S}_{++}^m\) (symmetric positive-definite matrices) is a Riemannian manifold of dimension:

A. \(m(m+1)/2\) — the number of independent entries in a symmetric matrix B. \(m\) C. \(m^2\) D. \(m(m-1)/2\)

Q10. The Levi-Civita connection on a Riemannian manifold is:

A. A physical connection between two manifolds B. An algorithm for computing geodesics C. The unique torsion-free connection compatible with the metric — it defines parallel transport and covariant differentiation in a way that preserves the Riemannian structure D. A type of coordinate chart