Curvature Bounds and Global Geometric Consequences
Let \((M,g)\) be a smooth Riemannian manifold, with Levi–Civita connection \(\nabla\). See the connection notes for the notation.
Curvature bounds constrain the global behaviour of geodesics. Positive curvature can produce shortening variations, while nonpositive curvature prevents a nontrivial Jacobi field starting at zero from returning to zero. We apply these two mechanisms to fundamental groups, global exponential coordinates, and diameter bounds.
Theorem 1 (Synge). A connected, compact, oriented Riemannian manifold without boundary, of even dimension and strictly positive sectional curvature, is simply connected.
Proof. Suppose a nontrivial free homotopy class of loops exists. We first justify the shortest-loop argument. Compactness gives a uniform positive radius on which short geodesics joining nearby points are unique and depend smoothly on their endpoints. Indeed, apply the inverse function theorem to \((x,v)\mapsto(x,\exp_xv)\) near the compact zero section. Sufficiently short loops lie in contractible coordinate neighborhoods, so lengths in our class have positive infimum \(\ell\). Take loops whose lengths decrease to \(\ell\), with constant-speed parametrizations on \([0,1]\). Their uniform Lipschitz bound and compactness give a uniformly convergent subsequence. Nearby loops are freely homotopic by short geodesic connectors. Thus the limit is in the same class; lower semicontinuity of length makes it a minimizer. Replacing short subarcs by their unique minimizing geodesics preserves the class, so equality forces the minimizer to be locally geodesic, also across the periodic joining point. We obtain a smooth nonconstant closed unit-speed geodesic \(\eta:[0,\ell]\to M\).
Parallel transport around \(\eta\) preserves the metric and orientation and fixes \(\eta'(0)=\eta'(\ell)\). Its restriction to the normal space is therefore \(P\in\mathrm{SO}(2d-1)\), where \(\dim M=2d\). An odd-dimensional orthogonal matrix with determinant \(1\) has eigenvalue \(1\): nonreal eigenvalues pair with their conjugates, while the product of the remaining odd number of real eigenvalues \(\pm1\) is \(1\). Parallel transport of a fixed unit vector gives a smooth periodic field \(Y\) with \(Y\perp\eta'\), \(|Y|=1\), and \(Y'=0\).
The variation \(c(s,t)=\exp_{\eta(t)}(sY(t))\) consists of closed loops in this free homotopy class. For \(E(s)=\frac12\int_0^\ell|\partial_tc|^2\,dt\), endpoint terms cancel by periodicity. Our curvature convention gives
Thus \(E(s)<E(0)=\ell/2\) for small nonzero \(s\). Cauchy–Schwarz gives \(L(c_s)^2\le2\ell E(s)<\ell^2\), a contradiction. There is no nontrivial free homotopy class, so \(\pi_1(M)=0\). In dimension zero, connectedness makes \(M\) a point. \(\square\)
Theorem 2 (Cartan-Hadamard). Let \(M\) be a complete, simply connected differentiable manifold with nonpositive sectional curvature, i.e. \(K(p,\sigma) \leq 0\) for all \(p \in M\) and \(\sigma \subset T_p M\). Then, \(M\) is diffeomorphic to \(\mathbb R^n\), where \(n = \dim M\), and \(\exp_p : T_p M \to M\) is a diffeomorphism.
The proof first excludes conjugate points, then equips \(T_pM\) with the pullback metric. Completeness makes the exponential map a covering local isometry, and simple connectivity makes this connected covering a diffeomorphism. An arbitrary immersion from a complete manifold need not be a covering map.
Proof. All manifolds here are connected and without boundary. Write \(T=\gamma'\). A nonzero Jacobi field with \(J(0)=0\) has \(J'(0)\ne0\) by uniqueness of its equation. Our curvature convention gives
Moreover \(|J(t)|^2=t^2|J'(0)|^2+O(t^3)\), so its derivative is positive for small positive \(t\). Convexity keeps it positive thereafter. Thus \(J(t)\ne0\) for every \(t>0\). The Jacobi differential formula makes \(\exp_p\) nonsingular away from \(0\); its differential at \(0\) is the identity.
Completeness of \(M\) defines \(\exp_p\) on all of \(T_pM\). The pullback \(h=(\exp_p)^*g\) is a positive definite smooth metric. Gauss's lemma says that \(r(v)=|v|_{g_p}\) satisfies \(|dr|_h=1\) away from \(0\). Integrating along curves, including across \(0\) by continuity, gives \(|r(v)-r(w)|\le d_h(v,w)\). Hence an \(h\)-Cauchy sequence lies in a bounded Euclidean ball and has a Euclidean-convergent subsequence. A smooth positive definite metric induces the Euclidean topology locally, so this subsequence converges also for \(d_h\). The Cauchy property then makes the full sequence converge. Thus \((T_pM,h)\) is complete.
We supply the covering argument. Let \(f:(X,h)\to(Y,g)\) be a local isometry with \(X\) complete and both manifolds connected. A finite-length piecewise smooth path in \(Y\) starting at \(f(x)\) lifts uniquely from \(x\): local inverses give the lift until a maximal time; near a finite endpoint its remaining length tends to zero, so the lift is Cauchy and converges in \(X\). A local inverse at its limit extends the lift. Any two points in connected \(Y\) can be joined by such a path, so \(f\) is surjective.
Choose a normal ball \(B(y,\varepsilon)\). For every \(x\in f^{-1}(y)\), lifting radial segments gives the smooth branch
Geodesic completeness of \(X\) defines this throughout the ball. The images of distinct branches are disjoint: if they met, lifting the reversed radial path uniquely would identify their initial points over \(y\). Reversed radial lifting also shows that the branches exhaust the preimage of the ball. Their images are open because each is locally an inverse of a local diffeomorphism. Thus the ball is evenly covered. Every geodesic in \(Y\) also extends for all time by lifting its initial data to \(X\), so \(Y\) is complete.
Apply this to \(\exp_p:(T_pM,h)\to M\). A connected covering of a simply connected manifold has one sheet, by path and homotopy lifting. Hence \(\exp_p\) is bijective, and its local smooth inverses give a global diffeomorphism. Identifying \(T_pM\) with \(\mathbb R^n\) finishes the proof. \(\square\)
1. Application: Estimates on Geometric Quantities
Theorem 3 (Bonnet–Myers). Let \(M\) be a complete connected Riemannian manifold without boundary of dimension \(d\ge2\). If \(\operatorname{Ric}(X,X)\ge\lambda|X|^2\) for all tangent vectors, where \(\lambda>0\), then
Moreover \(M\) is compact and \(\pi_1(M)\) is finite.
Proof. Choose distinct points \(p,q\) and a unit-speed minimizing geodesic \(\gamma:[0,\rho]\to M\), where \(\rho=d(p,q)\). Complete \(T=\gamma'\) to a parallel orthonormal frame \(T,X_2,\ldots,X_d\), and set \(Y_i(t)=\sin(\pi t/\rho)X_i(t)\). These fields vanish at both endpoints. Minimality implies \(I(Y_i,Y_i)\ge0\), even if the endpoint is conjugate. Since \(Y_i'=(\pi/\rho)\cos(\pi t/\rho)X_i\),
The \(X_i\), rather than \(Y_i\), are orthonormal. Integrating sine squared and cosine squared gives \(\rho/2\) for each, hence
This bounds every distance and proves the diameter estimate. Hopf–Rinow makes the complete bounded manifold compact.
The connected universal cover with its lifted metric is complete: a geodesic projects to one in \(M\), which extends for all time and whose extension lifts from the prescribed initial point. This local isometry preserves curvature, so the same diameter estimate makes the cover compact. Each covering fiber is closed and discrete, hence finite in this compact space. The fiber of a universal cover is in bijection with \(\pi_1(M)\), so the group is finite. \(\square\)
Proposition 4. Let \(M\) be a complete, connected, simply connected Riemannian manifold with \(K_M\le0\). Let \(\gamma:[0,\infty)\to M\) be a unit-speed geodesic ray, and let \(p\in M\). Here a ray is globally minimizing: \(d(\gamma(s),\gamma(t))=|s-t|\). There exists a unit-speed geodesic ray \(\lambda:[0,\infty)\to M\) with \(\lambda(0)=p\) and
The simple-connectivity hypothesis makes this a statement on a Hadamard manifold; it is used for global distance comparison and minimality.
The same limiting initial vector works for all times: smooth dependence gives convergence uniformly on each compact time interval. No separate subsequence for each \(t\) is required.
Proof of Proposition 4. Let \(d_0 = d_M(p, \gamma(0))\). Let \(\pi: \widetilde{M} \to M\) be the universal covering map. Equipped with the pullback metric, \(\widetilde{M}\) is a complete, simply connected Riemannian manifold with \(K_{\widetilde{M}} \le 0\). Thus, by the Cartan-Hadamard theorem (Theorem 2), \(\widetilde{M}\) is a Hadamard manifold and uniquely globally geodesic CAT(0) space. Because \(M\) is simply connected, this connected covering is a diffeomorphism. With the pullback metric it is a global isometry. In particular, it is distance non-increasing: \(d_M(\pi(x), \pi(y)) \le d_{\widetilde{M}}(x, y)\).
Lift the given ray \(\gamma\) to a geodesic ray \(\widetilde{\gamma}: [0, \infty) \to \widetilde{M}\). Choose a lift of \(p\), denoted \(\widetilde{p} \in \widetilde{M}\), such that the distance realizes the minimum in the fiber: \(d_{\widetilde{M}}(\widetilde{p}, \widetilde{\gamma}(0)) = d_M(p, \gamma(0)) = d_0\).
For each integer \(i>d_0\), let \(\widetilde{\lambda}_i : [0, L_i] \to \widetilde{M}\) be the unique unit-speed minimizing geodesic from \(\widetilde{p}\) to \(\widetilde{\gamma}(i)\), where \(L_i = d_{\widetilde{M}}(\widetilde{p}, \widetilde{\gamma}(i))\). By the standard triangle inequality on the vertices \(\widetilde{p}, \widetilde{\gamma}(0), \widetilde{\gamma}(i)\), we bound the length:
In a CAT(0) space, the distance function between two geodesics parameterized proportionally to arc length is convex. It need not be strictly convex: parallel Euclidean geodesics have constant separation. Consider the geodesics parameterized by proportion \(s \in [0, 1]\), defined as \(s \mapsto \widetilde{\lambda}_i(s L_i)\) and \(s \mapsto \widetilde{\gamma}(si)\). By convexity, we have
For fixed \(t>0\), choose \(i>t+d_0\). Then \(L_i\ge i-d_0>t\), so both evaluation times below lie in the segment domain. Set \(s=t/i\in(0,1)\); the convexity inequality gives
Next, we evaluate the exact distance \(d_{\widetilde{M}}(\widetilde{\lambda}_i(t), \widetilde{\gamma}(t))\) using the triangle inequality:
Because \(\widetilde{\lambda}_i\) has unit speed, the first term on the right-hand side is precisely the absolute difference of the arguments:
Combining these bounds, we obtain an upper bound for the distance at the same time \(t\):
We project this metric structure back down to \(M\) by defining \(\lambda_i = \pi \circ \widetilde{\lambda}_i\). These are unit-speed geodesics from \(p\) to \(\gamma(i)\) in \(M\). Because \(\pi\) does not increase distances:
The initial unit velocity vectors \(v_i = \lambda_i'(0)\) reside in the compact unit tangent sphere \(S_p M \subset T_p M\). We can pass to a convergent subsequence \(v_{i_k} \to v \in S_p M\). Let \(\lambda(t) = \exp_p(tv)\) be the unique unit-speed geodesic from \(p\) with initial velocity \(v\). Because geodesic flows depend smoothly and continuously on initial conditions, \(\lambda_{i_k}(t) \to \lambda(t)\) for all \(t \ge 0\). Taking the limit as \(k \to \infty\) in the distance bound yields:
It remains to check global minimality, which does not follow merely from being a geodesic. Fix \(0\le s\le t\). For all sufficiently large \(k\), the segment \(\widetilde\lambda_{i_k}\) is defined up to \(t\) and is minimizing. Since the covering is a global isometry in the present simply connected setting,
Passing to the limit gives \(d_M(\lambda(s),\lambda(t))=t-s\). Thus \(\lambda\) is a ray starting at \(p\), as required. ◻
The global comparison input used here is the convexity of distance between geodesics in a Hadamard space; see Ballmann, Lectures on Spaces of Nonpositive Curvature, Proposition I.5.4. Cartan–Hadamard supplies this global setting from the hypotheses.
Reading: Carmo (1992), Riemannian geometry.