The exponential map exp_p : TpM → M is defined by exp_p(v) = γ_v(1), where γ_v is the geodesic with γ_v(0) = p and γ_v'(0) = v. What does exp_p(tv) equal for t ∈ [0,1]?
AThe point at parameter t on the geodesic from p with initial velocity v
BThe point t · exp_p(v) (scalar multiplication in M)
CThe parallel transport of v along the geodesic for time t
DThe Riemannian exponential e^{tv} of the matrix v
By the scaling property of geodesics, the geodesic with initial velocity tv is a reparametrization of the geodesic with initial velocity v: γ_{tv}(1) = γ_v(t). So exp_p(tv) = γ_v(t), which is the point reached by following the geodesic from p with velocity v for time t. The curve t ↦ exp_p(tv) traces out the geodesic ray from p in the direction v. Option D refers to the matrix exponential, which motivates the name — on a Lie group, the two notions coincide.
Question 2 True / False
The exponential map exp_p is a diffeomorphism from all of TpM onto M.
TTrue
FFalse
Answer: False
The exponential map is only guaranteed to be a local diffeomorphism near the origin 0 ∈ TpM (by the inverse function theorem, since d(exp_p)_0 = id). Globally, it can fail to be injective (geodesics from p may intersect at other points, like antipodal points on a sphere) or fail to be defined for all v (if geodesics are not complete). The injectivity radius at p is the largest radius for which exp_p is a diffeomorphism on the open ball of that radius in TpM.
Question 3 Short Answer
What is the geometric significance of the injectivity radius of a Riemannian manifold?
Think about your answer, then reveal below.
Model answer: The injectivity radius inj(p) at a point p is the supremum of radii r such that exp_p is a diffeomorphism on the ball B_r(0) ⊂ TpM. Within this radius, every point has a unique minimizing geodesic from p, and normal coordinates are valid. The injectivity radius of M is inj(M) = inf_p inj(p). A positive injectivity radius guarantees that the manifold has uniformly 'Euclidean-like' neighborhoods. The injectivity radius is bounded below by curvature: Klingenberg's theorem gives inj(M) ≥ π/√K_max for even-dimensional manifolds with sectional curvature ≤ K_max.
The injectivity radius controls how 'locally Euclidean' the manifold is from the perspective of geodesics. Small injectivity radius means geodesics refocus quickly (due to positive curvature or topological complexity), making the manifold geometrically 'small.' Many theorems in Riemannian geometry require lower bounds on the injectivity radius to ensure analytic estimates work.
Question 4 True / False
In normal coordinates at p, the Riemannian metric satisfies gij(p) = δij and ∂kgij(p) = 0.
TTrue
FFalse
Answer: True
Normal coordinates are defined via the exponential map: the coordinate of exp_p(vⁱeᵢ) is (v¹,...,vⁿ). At the origin (the point p), the metric is the identity (because d(exp_p)_0 = id maps the standard inner product on TpM to itself), and the first derivatives of the metric vanish (equivalently, all Christoffel symbols vanish at p). The first nonzero correction to gij = δij is at second order and is controlled by the curvature: gij(x) = δij - ⅓Rikjl xᵏxˡ + O(|x|³).