Distance Functions, Curvature and Topology

In this note we collect and organize in detail, discuss and revisit some material about the properties of the distance functions on Riemannian manifolds, scattered throughout the three editions of the book of Peter Petersen [12,13,14], in order to connect their behavior with the geometry of the manifolds and possibly have a point of view, different from the "standard" one based on the study of the geodesics, leading to alternative proofs of some "classical" theorems connecting curvature and topology. Heuristically (having in mind the context of the calculus of variations), focusing on the geodesic flow is considering the "Lagrangian" point of view in studying the length (or energy) functional, since they are its critical points (solutions of its Euler equations, which are a system of ODEs), while considering the distance functions between points or subsets of the manifold is taking the "Eulerian" point of view, as the distance functions are the "value functions" associated to the length functional (using a terminology from optimal control theory, see [3] for instance). The straightforward connection between the two approaches is clearly given by the fact that the distance functions are realized by the length of the minimal geodesics and conversely, the geodesics are locally the integral curves of the gradients of the distance functions.

This connection also offers the possibility of using more “analytic” tools, related to the general study of the distance functions, seeing them simply as solutions of the Hamilton-Jacobi (eikonal) equation ∥∇u∥ = 1 (see [10]). Following Petersen, we describe this approach in general in the next section, then we specialize it to the particular distance functions from a fixed point (in Section 3), in order to connect the curvature and the topology/geometry of the manifold. The formulas we get, give us a set of information that can substitute what can be inferred about the behavior of geodesics, in the “classical” approach, using the second variation of the length functional and the theory of Jacobi fields.

As examples of application, we will show in Section 4 how these informations can be used to obtain alternative proofs (possibly easier or more intuitive than the “standard” ones) of the existence of local isometries between Riemannian manifolds of equal constant sectional curvature, of the Bonnet-Myers and Cartan-Hadamard theorems, of Synge’s lemma and in estimating the volume growth of geodesic balls.

Finally, it is also worth underlining that the approach to these results, which belong to the field called comparison geometry, is to describe locally the manifolds in polar coordinates and to “compare” the induced metric and the curvature of the geodesic spheres (not geometrically, but analitically, by studying the ODEs satisfied by their components along geodesics) with the analogous ones of appropriate space forms.

In the whole paper, we will consider only complete n-dimensional Riemannian manifolds with n ⩾ 2 and ∇ will be the Levi-Civita connection. Moreover, we will always use the Einstein convention of summing over repeated indices.

operator and tensor of (Σ ρ , g ρ ), respectively and if there is no risk of ambiguity, we will drop the superscript ρ.

We adopt the convention that

Riem(X, Y, Z, W ) = g(Rm(X, Y )Z, W ), while the sectional curvature of a plane generated by two linearly independent vectors v, w ∈ T p M is given by

.

We define S : Γ(T U ) → Γ(T U ) as

for every vector field X on U . It is the (1, 1)-version of the Hessian of r, indeed,

By means of tensor S, we can define the (1, 1)-tensor S 2 as S • S and the (0, 2)-tensor Hess 2 r by

The latter is clearly semidefinite positive. We also notice that in a local chart, there hold

, where ∂ i are the coordinate vector fields.

The Lie derivative of a tensor will be denoted by L, in particular, if X, Y are two vector fields, we have L

We then underline the relation

It says that the difference between the covariant and the Lie derivative with respect to ∂ r of a vector field X on U , is given by the tensor S applied to X.

Lemma 2.2. Let r : U → R be a distance function. Then we have

Hence, S(∂ r ) = 0, that is, ∂ r is a null eigenvector of S and Hess r(∂ r , •) = 0.

Proof. For every vector field X on U , we have

where we used the symmetry of the Hessian and g(∂ r , ∂ r ) = 1. □

A direct consequence of Lemma 2.2 is that any integral curve for ∂ r is a geodesic of (M, g), parametrized in arclength.

Remark 2.3. Obviously, by definition, we also have L ∂r ∂ r = 0.

The field ∂ r is the outward pointing unit normal vector field along any nonempty Σ ρ and, for every X ∈ Γ(T Σ ρ ), the vector field S(X) along Σ ρ belongs precisely to Γ(T Σ ρ ). Then, the shape operator of Σ ρ coincides with the map that associates S(X) ∈ Γ(T Σ ρ ) to each X ∈ Γ(T Σ ρ ), therefore it will be still denoted by S. The associated symmetric (0, 2)-tensor on the hypersurface Σ ρ , defined as/satisfying

for every pair of ta

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