Remarks on manifolds with two sided curvature bounds

We discuss folklore statements about distance functions in manifolds with two sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.

1. Introduction 1.1. Distance functions in smooth Riemannian manifolds. We discuss slightly generalized versions of some folklore results about distance functions d A to subsets A of smooth Riemannian manifolds M. The results turn out to be local and are proved without any completeness assumptions. Our proofs do not involve Jacobi fields, but only basic facts about semi-concavity and semi-convexity of distance functions. Therefore, the statements generalize to the synthetic setting of manifolds with two sided curvature bounds, as will be explained below.
The first statement is well-known in the complete situation, see, for instance, [MM03], [CS04], [ (1) f is semiconvex in O.
(3) f is C 1 in O.
(4) For any y ∈ O, there exists at most one geodesic γ y : [0, ǫ) → M starting at y and parametrized by arclength, with the property f • γ y (t) = f (y) − t, for all t in [0, ǫ).
We assume some familiarity with the notions of semiconcave (semiconvex) functions and their gradient flows, [Pet07], [AKP19], [AGS05]. Here and below, the notions of semi-concavity (convexity) and C 1,1 are local. For instance, we say that a function f is C 1,1 if it is C 1 and the gradient ∇f is locally Lipschitz continuous.
For a closed subset A of M, denote by Reg d A the set of points x ∈ M \ A, such that d A is C 1 in a neighborhood of x. Thus, Reg If M is complete, then the geodesic γ y in (4) can be extended to a unique minimizing geodesic γ y : [0, d A (y)] → M from y to A. The endpoint of this geodesic is the unique projection point Π A (y) of y on A. Moreover, the whole geodesic γ y ([0, d A (y))) is contained in Reg d A .
If M is complete and A is a C k submanifold, for k ≥ 2, then d A : Reg d A → R can be expressed in terms of the normal exponential map of A and turns out to be C k , see, for instance, [MM03,Prop. 4.3].
In the non-complete situation, none of the above statements need to hold, see Example 4.3.
The next result is known to specialists for some subsets in complete manifolds, [ACNS13], [Alb15]. We inlcude a short proof based on a general fact about gradient flows. Note, that the function d A is semiconcave on M \ A and has a uniquely defined (local) gradient flow. A natural generalization of the above result and its proof is valid in Alexandrov spaces, [AKP19, Prop. 14.1.5]. As an application of Proposition 1.2 one can derive a very short proof of the nice geometric observation [GS19, Theorem 6.1], see Corollary 4.2 below.
Recall that a closed subset A in a Riemannian manifold is said to have positive reach, if A has a neighborhood O such that the foot point projection Π A is uniquely defined on O. As has been shown by Federer and Bangert, the notion only depends on the underlying smooth structure and not on the Riemannian metric. The following result is essentially contained in [Kle81], [Ban82], [Lyt05b]. Proposition 1.3. Let A be a closed subset of a smooth Riemannian manifold M. Then the following are equivalent: See [Ban82] and [Lyt05b] for other characterizations. Any subset of positive reach A ⊂ M has a well-defined tangent cone T x A at every point x ∈ A. This tangent cone is a convex cone in T x M, [Fed69,Theorem 4.8]. The normal cone T ⊥ x A is the convex cone of all vectors in T x M enclosing angles at least π 2 with all vectors in T x A. For the following folklore statement about subsets of positive reach we could not find appropriate references. In the Euclidean case the result is contained in [Fed69].
1.2. Manifolds with two sided curvature bounds. Here we discuss some basic observations about non-smooth Riemannian manifolds which have two sided curvature bounds in the sense of Alexandrov and the extensions of the above results to this setting. These results have been applied in [KL20]. Readers only interested in the smooth situation can skip this part of the introduction together with Section 6.
Manifolds with two sided curvature bounds in the sense of Alexandrov appear in one of the most prominent examples of Gromov-Hausdorff convergence. Namely, the class of such compact manifolds with uniform bounds on injectivity radius, diameter and curvature is compact with respect to the Gromov-Haudorff convergence, [BN93], [Gro07], [Pet87]. It provides a natural compactification of the corresponding class of smooth Riemannian manifolds. Moreover, all manifolds with two sided curvature bounds in the sense of Alexandrov turn out to have a rather regular analytic structure and to admit a smoothing, as has been proved by Nikolaev in a series of papers. A good readable summary of these results of Nikolaev and related statements on the structure of such manifolds has appeared in [BN93]. Results and ideas of Nikolaev motivated many theorems in the theory of Alexandrov spaces with one-sided curvature bounds, [BGP92], [KKK19], [Per94], [OS94], [Pet98], [LN18].
We will assume some familiarity with the theory of Alexandrov spaces, and refer the non-familiar reader to [AKP19]. The most appropriate setting for our local results is the following one.
Definition 1.5. A locally compact length metric space X has two sided bounded curvature if for any point x ∈ X there exists a compact convex neighborhood U and some K > 0 such that U is an Alexandrov space of curvature ≥ −K and a CAT (K) space.
A space X with two sided bounded curvature is topologically a manifold M with boundary ∂M. Moreover, M \∂M is convex in M, [BN93], cf. [KKK19]. We restrict the attention to the case ∂M = ∅.
By a manifold with two sided curvature bounds we will denote a space as in the above Definition 1.5 which, in addition, is homeomorphic to a manifold without boundary.
Any manifold M with two sided curvature bounds admits a natural atlas of distance coordinates, see [BN93] and Section 3 below. The distance in M is defined by a C 0,1 Riemannian metric g in this atlas. Moreover, the C 1,1 -smoothness of the atlas and the C 0,1 -smoothness of the Riemannian metric is optimal, as can be observed in the manifold M arising from the gluing of a flat cylinder and a hemisphere. Results of this type with weaker conclusions have been obtained for distance coordinates under one-sided curvature bounds, [Per94], [OS94], [LN18].
On any C 1,1 manifold M with a Lipschitz continuous Riemannian metric, any harmonic function is C 1,α , for all α < 1, [Tay00, III, Chapter 9] and there exist harmonic coordinates around any point, [Tay00, III, Chapter 9]. The atlas of harmonic coordinates is of class C 2,α , [Sv76, p. 689] and the distance is defined by a metric of class C α in these coordinates. One of the central results of Nikolaev's theory, see [BN93], is that for any manifold M with two sided curvature bounds, the harmonic atlas is of class C 3,α , for any α < 1, and the Riemannian metric in this atlas is of class C 1,α , for any α < 1.
In a general C 1,1 -manifold with a C 0,1 Riemannian metric a harmonic function does not need to be of class C 1,1 , cf. [Sv76, p. 693] and Problem 1.9 below. Our first observation is that such a loss of smoothness cannot happen on a manifold with two sided bounded curvature.
Proposition 1.6. Let M be a manifold with two sided bounded curvature. Then any harmonic function on an open subset U in M is of class C 1,1 in distance coordinates. Thus, any transformation from distance to harmonic coordinates is of class C 1,1 .
Thus, the C 1,1 atlas of distance coordinates can be assumed to include all harmonic coordinates as well. From now on all statements will refer to this C 1,1 atlas. The following result might be folklore knowledge: Proposition 1.7. Let M be a manifold with two sided curvature bounds and let N ⊂ M be a C 1,1 submanifold. Then N with its intrinsic metric is a manifold with two sided curvature bounds.
This result follows from the Gauß equation and another central result of Nikolaev's theory, stating that manifolds with two sided curvature bounds are exactly the limit spaces (in a precise local sense) of smooth Riemannian manifolds with uniform bounds on sectional curvature.
The final statement discussed in the introduction is that all results about distance functions in smooth Riemannian manifolds are valid in this more general setting: Theorem 1.8. The statements of Proposition 1.1, Proposition 1.2 and Proposition 1.3 are valid for any closed subset A of any manifold M with two sided bounded curvature. The differentiability in Propositions 1.1, 1.3 is considered with respect to distance coordinates.

A few questions.
We would like to finish the introduction with a few open questions about manifolds with two sided curvature bounds. Problem 1.9. Does there exist a Riemannian metric of class C 0,1 and a harmonic function with respect to this metric, which is not C 1,1 ?
It is possible that a positive answer to this problem might be obtained following the ideas in the examples discussed in [Sv76,p. 693].
The second part of the next problem is motivated by Proposition 1.7. It should be compared with Nash's embedding theorem for Riemannian manifolds of higher regularity, see [And02].
Problem 1.10. Let M be a (compact, complete, local) manifold with two sided curvature bounds in the sense of Alexandrov. Do there exist coordinates on M in which the distance is defined by a Riemannian metric of class C 1,1 ? Can M be length-preserving embedded as a C 1,1 submanifold in a Euclidean space?
We would like to mention that the folklore argument providing a negative answer to the first question above, for instance, [Pet87], is not correct. Indeed, [Pet87] provides a surface with two sided curvature bounds such that in harmonic coordinates the Riemannian metric is not of class C 1,1 . Then the proof invokes the statement of [DK81], [Sv76] that the smoothness of the metric is optimal in harmonic coordinates. However, this elliptic regularity statement is not covered by [DK81] (since elliptic regularity does not work well for Lipschitz functions). Indeed, the following result has appeared in [Sv76, Example 2] and provides a counter-example to the folklore proof. There exists a Riemannian metric of class C 2 on R 2 such that in any harmonic coordinates the metric is not C 1,1 .
The second question concerns the existence of a canonical smoothing of manifolds with two sided curvature bounds. While the existence part in the next question is a direct consequence of the approximation results of Nikolaev and [Shi89], the uniqueness is more subtle: The final problem we would like to mention is due to the following fact. The theory of Nikolaev is scattered through several works and some of them are not easy to read (and to find). Thus, we formulate: Problem 1.12. Find a streamlined proof of Nikolaev's result on smoothings of manifolds with two sided curvature bounds, [Nik91], [Nik88].
1.4. Structure of the paper. In Section 2, we recall basics on semiconcave functions and their gradient flows and verify a local version of Proposition 1.2. In Section 3, we recall basic facts about distance coordinates. In Section 4, we prove Proposition 1.1 and 1.2. In Section 5, we prove the stated results about subsets of positive reach, Propositions 1.3, 1.4. Finally, in Section 6, we prove Proposition 1.6 and Proposition 1.7.
2. Semiconcavity and gradient flows 2.1. Notation. Distance will be denote by d. The distance function from a subset A of a space X will be denote by d A . By definition, this is a 1-Lipschitz function. A geodesic will denote an isometric embedding γ : I → X of an interval. Thus, our geodesics are always parametrized by arclength and globally minimizing.
2.2. Special neighborhood. Let M be a manifold with two sided curvature bounds. For any point x ∈ M, we find a compact neighborhood U of x as in Definition 1.5. Restricting the neighborhood and using convexity of small balls in CAT (K) spaces we may assume that the neighborhood U = U x has the following form.
The set U x is the closed ball of radius r x around x and it is homeomorphic to a Euclidean ball, [BN93, Theorem 12.1]. Any pair of points in U x is connected by a unique geodesic in U x . For some K x > 0, the space U x is CAT (K x ) and an Alexandrov space of curvature ≥ −K x , moreover, ǫ x := r x · K x << 1. Any geodesic in U x extends to a geodesic starting and ending on the distance sphere ∂U A function f is C-concave if and only if for all pairs of points p 1 , p 2 ∈ M which are sufficiently close to each other and any midpoint m between p 1 and p 2 , we have .
This is equivalent to the requirement that, for any geodesic γ : I → O, we have (f • γ) ′′ ≤ h • γ on I in the sense of distributions.
Since the notion of semiconcavity and of gradient curves and flows of semiconcave functions is local, the whole theory of gradient flows in Alexandrov spaces, [AKP19], [Pet07], applies to the present situation.
For any semiconcave function f : O → R and every x ∈ O there exists a unique vector ∇ x f ∈ T x O, the gradient of f at x. Moreover, there exists a unique maximal curve η x : [0, a) → O with some a ∈ (0, ∞), the gradient curve of f , which starts in x and satisfies In this case η x is the stationary curve η x (t) = x. On the other hand, if the curve η x does not contain critical points of f , it has a unique parametrizatioñ η x : [0,ã) → O by arclength.
The following is a not very well-known but fundamental observation. The first statement is exactly [AKP19, Theorem 14.1.3]; the second one follows from the first by localization: 2.4. Distance functions in special neighborhoods. Let U ⊂ M be a special neighborhood of some point as in Subsection 2.2. Thus, U is convex, compact, CAT (K) and it is an Alexandrov space of curvature ≥ −K, for some K > 0.
Then, for any point x ∈ U the distance function f = d x is convex in U. Moreover, f is also semiconcave on U \ {x}. More precisely, on the set O of points y ∈ U with d(x, y) > δ, the function f is C-concave, for some C = C(K, δ).
Since an infimum of C-concave functions is C-concave, for any subset B ⊂ U the distance function d B is C(K, δ) concave on the set of all points y ∈ U with d B (y) > δ.
The function f = d B is 1-Lipschitz, thus |∇ y f | ≤ 1 for all y ∈ U \ B. By the first variation formula, |∇ y f | = 1 if and only if y is connected with B by a unique shortest geodesic.
The conclusion of Proposition 1.2 will easily follow from the next Lemma, cf. [AKP19, Prop. 14.1.5], [ACNS13, Theorem 4.5]: Lemma 2.2. Let B ⊂ U be any closed subset in a special neighborhood U as above. Let y ∈ U \ B be arbitrary and let η y : [0, a) → U be the gradient curve in U of the distance function f = d B starting at y. If |∇ y f | < 1 then |∇ z f | < 1, for all z on the gradient curve η y .
Proof. Rescaling the space, we may assume that the lower curvature bound on U equals −1. Then, for the distance function F = d p to any p ∈ U the compositionF = cosh •F isF -concave on U, [AKP19, Theorem 7.4.1]. Hence, also for the infimum f = d B of distance functions to points, the compositionf := cosh •f isf -concave on U.
The gradient curves of f andf on U \ B coincide up to parametrizations, [AKP19, Theorem 11.4.4]. Thus, the arclength reparametrizationη y of η y is also the arclength parametrization of the gradient curve off . From Lemma 2.1, we deduce thatf •η y isf •η y -concave on the interval of definition [0, b] ofη y from y to z.

Assume, by contrary that h ′ (b) = 1. Consider the linear function
This implis h ′ (0) = 1 in contradiction to our assumption. This finishes the proof.
2.5. Distance functions to remote subsets. Let M be a manifold with two sided curvature bounds (possibly non-complete and without uniform bounds on curvature), as in Subsection 1.2. Let A be a closed subset of M and consider the distance function f = d A to A.
Consider an arbitrary point x ∈ M \ A and a special neighborhood U = U x =B rx (x) as in Subsection 2.2. Making r x smaller, we may assume that U is disjoint from A.
LetŨ be a smaller ball of radius r < 1 3 ·r x around x. Set s = f (x) and denote by B the compact subset B = f −1 (s − 2r) ∩ U. By compactness and convexity of U, for any y ∈Ũ , we find at least one footpointŷ of y on B, thus d(y,ŷ) = d B (y). By the triangle inequality, On the other hand, any curve η from A to y must contain points z with f (z) = s − 2r. If z is not in U then the length of η is at least s + r. Thus, any such η with length less than s + r, contains points on B. Thus by the triangle inequality we deduce the equality onŨ. Since d B is semiconcave onŨ , so is f , moreover, the gradient curves of f and d B coincide inŨ. Now we arrive at the following results (also valid in all Alexandrov spaces and their localized version, Alexandrov regions, [LN20]). Proof. The semiconcavity condition is local and has been verified above in the neighborhood of any x ∈ M \ A.
In order to see the second statement, we consider the compact part η x : [0, b] → M of the gradient curve η x between x and y. Assuming the contrary, and using that f •η x is semiconcave, we find a smallest t ∈ (0, b] such that |∇ ηx(t) f | = 1. Now we find a small special neighborhood U of z = η x (t), identify on a smaller neighborhoodŨ (up to an additive constant) f with d B for a closed subset B ⊂ U and derive a contradiction to Lemma 2.2. Proof. From the biLipschitz porperty of F , and semiconcavity and semiconvexity of the coordinates f i , we deduce the following. For any geodesic γ in O connecting q 1 and q 2 and having m as its midpoint the distance between F (m) and the midpointm = 1 2 (F (q 1 ) + F (q 2 )) in R n between F (q i ), we have Here the constant C depends only on the biLipschitz constant of F and the curvature bounds.
Since F is biLipschitz, F −1 sends midpoints inÕ to "almost midpoints" in O in the same sense as above. Now the equivalence of semiconcavity of f and f • F follows after applying (2.1).

Main results
4.1. General distance functions. We are going to prove the following slight generalization of Proposition 1.1 (1) f is semiconvex in O.
( The semiconcavity of f has been verified in Corollary 2.3. Since on open subsets of R n a function is C 1,1 if and only if it is semiconcave and semiconvex, the properties (1) and (2) are equivalent.
By the first formula of variation, |∇ x f | = 1 if and only x is connected with B by exactly one shortest geodesic γ x . Note that for any such Moreover, for any geodesic,γ x : [0, ǫ) → U satisfying the above equality for all t ∈ [0, ǫ), the unique extension ofγ x to length f (x) ends on B. This shows the equivalence of (4) and (5).
It remains to show that (4) implies (1). In this case, gradient lines η x of f in O are parametrized by arclength and f • η x has everywhere velocity 1. This implies that any gradient line η x in O is an (as always minimizing) geodesic. Fix x ∈ O and consider δ such that the ball of radius 3δ around x is contained in O. Consider C = C(K, δ), the semiconcavity constant of distance functions in O at distance ≥ δ, as in Subsection 2.4. We claim that f is −C-convex on the ball W of radius δ around x.
Indeed, consider points q 1 , q 2 ∈ W and their midpoint m. Set z = η m (2δ). Then, for i = 1, 2, due to the triangle inequality. This implies This implies the claimed semiconvexity of f .

Cut locus. As a combination of previous results we now obtain
Proof of Proposition 1.2. As before set f = d A . Due to Proposition 4.1, the cut locus CL(A) of A is exactly the closure CL(A) =X of As we have seen in Corollary 2.3, the set X is invariant under the gradient flow of f . By continuity of the gradient flow, the closurē C = CL(A) is invariant under the gradient flow as well.
As a consequence we deduce the following observation from [GS19]: Proof. Assume the contrary, thus d A (x 0 ) < d A (x). Consider a geodesic γ from x 0 to x. Since h := d A • γ is concave, the derivative of h at 0 is positive. In particular, x 0 is non-critical for the function d A . Thus, the gradient curve η x 0 of d A starting at x 0 is non-constant. Moreover, the angle between γ ′ (0) and ∇ x 0 d A is less than π 2 .
By the first variation formula, the derivative of l(t) := d(x, η x 0 (t)) at 0 is negative. Hence, for small t > 0, we have Due to Proposition 1.2, η x 0 (t) is contained CL(A). Thus, x 0 is not a nearest point to x in CL(A). This contradiction finishes the proof. 4.3. Simple counterexample. The following example shows that without a completeness assumpion no higher smoothness of the distance function can be expected: It remains to show that (2) implies (3). Thus, let O be a neighborhood of A such that f = d A is C 1,1 in O \ A. Then f is semiconvex on O \ A and we need to show that f is semiconvex around any point x ∈ A. Consider a special ball U x around x of radius r. Let W be the ball of radius r 3 around x. For any q 1 , q 2 ∈ W with midpoint m we either have m ∈ A and then Or m / ∈ A. Then the gradient curve η m of f starting in m is a geodesic on [0, r 3 ]. As in the last part of the proof of Proposition 1.1, we deduce for some C depending only on U x . Hence f is semiconvex in W , finishing the proof.

Geodesics in normal directions. Now we derive:
Proof of Proposition 1.4. The distance function f = d A is semiconvex in a neighborhood O of A. Thus, we have a well-defined (directional) differential D x f : T x M → R and this differential coincides with the distance to the tangent cone T x A, cf. [Lyt05a]. By the definition of normal directions, we have D x f (h) = 1. In other word, By the first variation formula, this implies that for t → 0, the angle between (γ h ) ′ (t) and the unique shortest geodesic from γ h (t) to A converges to π. Thus the angle between the gradient curve η t of f starting at γ h (t) and γ h at the point γ h (t) converges to 0. Since, the gradient curves η t are geodesics on a fixed interval [0, s 0 ], we deduce that Due to Nikolaev's theorem, the pull-back Riemannian metricg := (F −1 ) * (g) is C 1,α onÕ := F (O) ⊂ R n , for all α < 1, [BN93, Theorem 14.2]. In particular,g is locally Lipschitz continuous.

Submanifolds. This subsection is devoted to
Proof of Proposition 1.7. Let M be a manifold with two sided bounded curvature and let N be a C 1,1 submanifold with respect to distance coordinates.
The statement is local. We fix any point x ∈ N and consider a special neighborhood U x of x in M. Then we find a compact C 1,1 submanifold N of M which is contained in U x and such that N andN coincide in a neighborhood of x.
Since the statement is local we may assume that N =N. We may further assume that U x is a distance chart and identify it with a ball in R n . Then the distance in U is given by a Lipschitz continuous Riemannian metric g.
Due to [Lyt05b, Proposition 1.5] and [Lyt04, Theorem 1.2], the space N is CAT (κ) for some κ. It thus remains to show that N has curvature bounded from below.
Applying Proposition 1.6 and Nikolaev's approximation theorem, [BN93,Theorem 15.1], we may further assume on U there exist smooth Riemannian metrics g ǫ , which are uniformly Lipschitz continuous and converge to the Riemannian metric g. Moreover, the Riemannian manifold (U, g ǫ ) have sectional curvature uniformly bounded from above and below.
Smoothing the submanifold N we find a family of smooth submanifolds N ǫ ⊂ U with uniform bounds on their C 1,1 -norms (thus any N ǫ is a union of a uniform number of charts each of them of bounded C 1,1 -norm), such that N ǫ converge to N in C 1 sense.
Considering N ǫ with the the intrinsic metric induced by g ǫ , we see that N ǫ converge to N in the Gromov-Hausdroff metric (in fact, the convergence is much stronger). Thus, it remains to show that all N ǫ are Alexandrov space of curvature ≥ −κ, for some fixed κ. However, the C 1,1 bounds of N ǫ directly imply, that the second fundamental forms of N ǫ are uniformly bounded (as subsets of the flat space R n and therefore, of the Riemannian manifold (U, g ǫ )). Applying the Gauß equation, we derive a uniform lower bound on the sectional curvatures of N ǫ . By Toponogov's theorem and compactness of N ǫ , the manifolds N ǫ are Alexandrov spaces of curvature ≥ −κ. This finishes the proof.