Some Regularity of submetries

We discuss regularity statements for equidistant decompositions of Riemannian manifolds and for the corresponding quotient spaces. We show that any stratum of the quotient space has curvature locally bounded from both sides.


Introduction
A submetry is a map P : X → Y between metric spaces which sends open balls B r (x) in X onto balls B r (P (x)) of the same radius in Y .Submetries as metric generalization of Riemannian submersions have been introduced by Berestovskii [Ber87].Berestovksii and Guijarro verified that a submetry between smooth complete Riemannian manifolds always is a C 1,1 Riemannian submersion, but it does not need to be C 2 [BG00].Another example of a submetry is provided by a distance function P : R n → R to a convex nowhere dense subset C ⊂ R n .
Submetries P : X → Y with given total space X are in one-to-one correspondence with equidistant decompositions of X.The correspondence assigns to P the decomposition of X into fibers of P [KL22, Section 2.2].Seen this way, submetries generalize quotient maps for isometric group actions and decompositions of a complete smooth Riemannian manifold into leaves of a singular Riemannian foliation with closed leaves [Mol88], [LT10], [AC22].
If the total space X is a connected (sufficiently smooth) complete Riemannian manifold M, the following structural results on the base space Y of a submetry P : M → Y have been derived in [KL22].
The quotient space Y locally has curvature bounded from below, [BGP92], [KL22, Proposition 3.1].There is a canonical stratification Y = ∪ m l=0 Y l , where m is the dimension of Y and Y l consists of all points y ∈ Y such that the tangent space T y Y has R l as a direct factor, [KL22,Theorem 1.6].The subset Y l is locally convex in Y , for any l, and it is an l-dimensional manifold.The maximal-dimensional stratum Y m , the set of regular points of Y , is open, dense and convex in Y .
For any point y ∈ Y there exists some r > 0, such that exponential map exp y is a well-defined homeomorphism exp y : B r (0) → B r (y) between the r-ball in the tangent cone T y Y around the origin and the r-ball in Y around y, [KL22,Theorem 1.3].This injectivity radius is locally bounded from below on each stratum Y l , but it goes to 0, when points on Y l converge to a lower-dimensional stratum.
Our first result improves the regularity of the exponential map: Theorem 1.1.Let Y be a base of a submetry P : M → Y of a Riemannian manifold with locally bounded curvature.Then, for any y ∈ Y , there exist r 0 , C > 0, such that for all r < r 0 the exponential map exp y : B r (0) → B r (y) is (1 + Cr 2 )-bilipschitz.
Here and below, we use the notion of a Riemannian manifold with locally bounded curvature to describe a manifold without boundary with a continuous Riemannian metric, which has curvature bounded locally from above and below in the sense of Alexandrov see [BN93], [KL21].Any C 1,1 -submanifold of a Riemannian manifold with a C 1,1 -Riemannian metric is in this class [KL21, Proposition 1.7].
Theorem 1.1 can be informally understood as the existence of a pointwise both-sided curvature bound at any point y ∈ Y .Indeed, for a smooth Riemannian manifold M = Y , the optimal number C in the statement of Theorem 1.1 is equivalent (up to a factor) to the optimal bound on the norm of the sectional curvatures at y.
In Theorem 1.1, the constant r 0 (y) always goes to 0 and C(y) usually goes to infinity, when y converges to a lower stratum, [KL22, Proposition 8.9], [LT10, Theorem 1.1].But both constants can be chosen locally uniformly on any stratum, Theorem 6.1 below.This has the following consequence, which answers [KL22, Question 1.12]: Corollary 1.2.Let M be a Riemannian manifold with locally bounded curvature and let P : M → Y be a submetry.Then, any stratum Y l of Y is a Riemannian manifold with locally bounded curvature.
For smooth Riemannian manifolds M the result will be strengthened in the continuation [LW23].If M is analytic, the analyticity of the maximal stratum Y m has been verified in [Lem19].
In general, fibers of a submetry P : M → Y can be arbitrary subsets of positive reach in M (this is a common generalization of convex subsets and C 1,1 submanifolds [Fed59], [Lyt05a], [RZ17]).However, most fibers are C 1,1 -submanifolds and any fiber L of P contains a C 1,1submanifold, open and dense in L. A by-product of the proof of Theorem 1.1 is the following result saying that for any submetry P : M → Y the projections from nearby P -fibers onto any manifold P -fiber is almost a submetry.We formulate it as a global result for compact fibers and refer to Theorem 5.2 for a more general local version.
Proposition 1.3.Let P : M → Y be a submetry, where M has locally bounded curvature.Let L be a fiber of P which is a compact manifold.Then there exist constants C, r 0 > 0 such that for all fibers L ′ of P at distance r < r 0 from L, the closest point projection Recall that a map f : X → Y between metric spaces is locally Copen, (other terms used are Lipschitz open and co-lipschitz ) if for any z ∈ X there exists r 0 > 0, such that, for any r < r 0 and any x ∈ B r 0 (z), A submetry P : M → Y is called transnormal if all fibers of P are C 1,1 -submanifolds.Thus, for transnormal submetries with compact fibers the conclusion of Proposition 1.3 is true for all fibers.Moreover, for transnormal submetries, the constants C, r 0 appearing in Proposition 1.3 and in Theorem 1.1 depend only on the following data: A bound on the curvature and the injectivity radius of M, a bound on the injectivity radius of Y at y = P (L) and a lower volume bound of Y around y, see Corollary 7.4 below.This seems to be useful for applications to the theory of Laplacian algebras developped by Ricardo Mendes and Marco Radeschi, [LMR23].
Theorem 1.1 implies that the local decomposition of the base space Y in strata around a point y corresponds to the decomposition in strata of the tangent space at y, see Corollary 6.3 below.This has the following consequence for transnormal submetries: Corollary 1.4.Let P : M → Y be a transnormal submetry, where M has locally bounded curvature.Let γ : I → M be a horizontal geodesic.Then, up to discretely many values t i ∈ I, the connected component of the fiber of P through γ(t) has the same dimension k = k(γ) and P (γ(t)) is contained in the stratum Y l , with l = l(γ).
As a related consequence of Proposition 1.3, we prove that all holonomy maps between fibers of transnormal submetries are Lipschitzopen, see Proposition 7.2 below.
We mention, that all results stated here and below do not require completeness of M and are valid for local submetries, see Section 3.1.
Acknoweldgements The author is grateful to Ricardo Mendes and Marco Radeschi for their interest and helpful comments.
2. Preliminaries: Manifolds with bounded curvature 2.1.Notation.By d we denote the distance in metric spaces.For a subset A of a metric space X we denote by d A : X → R the distance function to A. A geodesic will denote an isometric (i.e.globally distance preserving) embedding of an interval.A local geodesic γ : I → X is a curve whose restrictions to small sub-intervals are geodesics.
2.2.Curvature bounds and bounds of geometry.We assume some familiarity with spaces with curvature bounded in the sense of Alexandrov.We refer the reader to [BN93], [AKP19].
By a manifold with locally bounded curvature M we mean a length metric space homeomorphic to a manifold without boundary, such that any point x ∈ M has a convex neighborhood in M, which is a CAT(κ) space and an Alexandrov space of curvature bounded from below by −κ, for some κ ∈ R. We allow the manifold M to be non-complete and the value κ to be not globally bounded on M.
The distance coordinates define a C 1,1 -atlas on any such manifold M and the Riemannian metric is Lipschitz continuous in these coordinates, [BN93].Any C 1,1 -submanifold N ⊂ M also has locally bounded curvature in its intrinsic metric [KL21, Proposition 1.7].
In any manifold M with locally bounded curvature there is a notion of parallel translation along any Lipschitz curve [BN93, Section 13].
Let x be a point in a manifold M with locally bounded curvature and let ρ > 0 be given.We say that the the geometry is bounded by ρ at x if the following conditions hold true: The ball B = B 10 ρ (x) is compact, convex and uniquely geodesic and the curvature in B is bounded from below and above by ± ρ 2 100 .If the geometry of M at x is bounded by ρ, then the metric space λ • M rescaled by λ > 0 has at x geometry bounded by ρ λ .Let the geometry of M at x be bounded by ρ and consider the ball B = B 1 ρ (x).Consider some distance coordinates on B and the Lipschitz continuous metric tensor g defining the metric of B in these coordinates.Then there exist a sequence of smooth, uniquely geodesic metrics g n on B such that the sectional curvatures of g n are bounded in norm by ρ 2 100 , with the following properties [BN93, Section 15]: The metric tensors g n converge to g in C 0,1 and the parallel transport for g n uniformly converges to the parallel transport for g.
This approximation result allows us to prove metric statements in the smooth case first and then to obtain the general case by a limiting procedure.Mostly, a more direct but technical explanation is available without using the approximation theorem.The main additional tool available in the smooth situation are Jacobi-fields, which only have almost everywhere analogues in the general case.Readers not aquainted with the theory of [BN93] may always assume the total manifold M to be smooth.We prefer to stick to the more general setting of manifolds with locally bounded curvature, since this setting seems to be appropriate for the study of submetries, see [KL22].This "quasi-distance" is symmetric but satifies the triangle inequality only up to a defect depending on the geometry of O, see (2.2) below.
For linear subspaces Here, the distance d(W x , w ′ ) to the subspace W x is measured in T x O, and the supremum is taken over all parallel translates w ′ ∈ W x of unit vectors w ∈ W z along the geodesic xz.

Almost flat domains.
We fix ε = 10 −4 for the rest of the paper.
We say that M is almost flat at x ∈ M if the geometry of M at x is bounded by ε.
If M has geometry bounded by ρ at x then, for any λ ≥ ρ ε , the rescaled manifold λ • M is almost flat at x.
Let M be almost flat as x and consider the open ball O = B 10 (x).Thus, O is convex and uniquely geodesic and the curvature in O is bounded from both sides by ± ε 2 100 = ±10 −10 .We refer to [BK81, Section 6] for the estimates stated below.For any ball B r (x) ⊂ O, the exponential map exp For such ∆ and arbitrary Then the holonomy bound (2.1) implies a triangle inequality with a defect: (2.2) The next result is a direct consequence of [BK81, Proposition 6.6].
Lemma 2.1.Assume B 4 (x) ⊂ O and u, w ∈ T x O with ||u||, ||w|| < 2. Set z = exp x (w) and p = exp x (u) and q = exp x (w + u).Let ũ be the parallel translate of u to z and q ′ := exp z (ũ).Then Therefore, And Corollary 2.2.Let w, u ∈ T x O be given with ||w|| = 1.Consider the curve η(t) := exp x (u + tw).Then, for all sufficiently small t, the starting direction w j of the geodesic connecting η(0) and η(t) satisfies We will need the following (definitely not optimal) lemma: Proof.Let h ∈ T x 0 O be the starting directions of γ.Let h be the parallel translation of h to z 0 .Set zt = exp z 0 (t • h) and ṽt = exp −1 xt (z t ).From (2.4) we deduce Applying (2.4) again, we deduce Together with (2.7) this implies 2.5.Subsets of positive reach.Let M be a Riemannian manifold with locally bounded curvature.A locally closed subset L ⊂ M has positive reach in M if the closest-point projection Π L in uniquely defined on a neighborhood U of L in M. In this case, Π L is locally Lipschitz on U and the distance function The following result is essentially contained in [Lyt05a, Theorem 1.6, Theorem 1.2].It formalizes the following observation: For a C 1,1 submanifold a lower bound on the reach is equivalent to an upper bound of the second fundamental form, as well as to a C 1,1 -bound of the submanifold.The proof consists just of a few citations.
Lemma 2.4.There exists c > 0 with the following properties.
Let the geometry of M at p be bounded by ρ.Let L be a closed subset containing p, such that the closest point projection Π = Π L onto L is uniquely defined in B 10 ρ (p).
Then, for r ≤ 3 ρ , the Lipschitz constant of Π on B r (p) is at most If, in addition, L is a C 1,1 -submanifold then, for all q ∈ L ∩ B r (p), (2.9) We find a constant c = c(C) with e Ct ≤ 1 + c • t, for all |t| ≤ 4. Since Φ(x, r) = Π L (x), for d(L, x) ≤ r, we obtain (2.8), for all 0 < r ≤ 4.
We turn now to (2.9).The existence of some constant c = c(L, M) satisfying (2.9) is equivalent to the property that L is a C 1,1 -submanifold.The claim is that c can be chosen independently of L and M We fix a sufficiently small (but universally chosen) 1 >> δ > 0 to be determined later.It is sufficient to prove (2.9) for all r < δ.
Since L ′ is a manifold, local geodesics in K starting in p ′ are extendable as local geodesics until the relative boundary of K in L ′ [LS07, Theorem 1.5].Moreover, these local geodesics are minimizing in K on intervals of length δ = δ(κ), due to the CAT(κ) property.Now a uniform Lipschitz estimate for the map

Basics on submetries
3.1.(Local) Submetries.Recall that P : X → Y is a submetry if for any x ∈ X and any r > 0 the equality P (B r (x)) = B r (P (x)) holds.
The map P is called a local submetry if for any x ∈ X there exists some s > 0 such that the condition P (B r (z)) = B r (P (z)) holds true for any z ∈ B s (x) and any r < s.We call X the total space and Y the base of the local submetry P .
P Let P : X → Y be a local submetry and let X be a length space.Replacing Y by P (Y ) we may assume that the local submetries are surjective.Replacing the metric on Y by the induced length metric, P remains a local submetry [KL22, Corollary 2.10].Thus, we may assume without loss of generality that the base space Y is a length space.

3.2.
Structure of the base.From now on let M denote a manifold with locally bounded curvature.Let P : M → Y be a surjective local submetry.Let y ∈ Y be arbitrary.
There exists some r = r(y) > 0, such that any geodesic γ : [0, t] → Y starting in y can be extended to a geodesic γ : [0, r] → Y up to the distance sphere ∂B r (y) [KL22, Theorem 1.3].In this case we will say that the injectivity radius at y is at least r.Under the above assumptions, any point y ′ ∈ B r (y) is connected to y by a unique geodesic. Set Here, Y l is the set of all points y ∈ Y , for which the tangent space T y Y splits off R l but not R l+1 as a direct factor.Y l is an l-dimensional manifold with a canonical C 1,1 -atlas, which is locally convex in Y , [KL22, Theorem 1.6].The metric on Y l is given by a Lipschitz continuous Riemannian metric; the tangent space T y Y l is the maximal Euclidean factor of T y Y [KL22, Theorem 11.1].
For any point y ∈ Y l , there exists some r 0 = r 0 (y) > 0 with the following properties [KL22, Lemma 10.1, Theorem 11.1]:The open ball B 2r 0 (y) does not contain points in ∪ l−1 i=0 Y i and, for any y ′ ∈ B r 0 (y) ∩ Y l , the injectivity radius at y ′ is at least r 0 .Neither L nor S have to be manifolds.However, for every y ∈ Y \∂Y the fiber

3.3.
In particular, this applies to all y ∈ Y m with m = dim(Y ).
3.4.Infinitesimal structure.Let P : M → Y be a local submetry, let x ∈ M be arbitrary, y = P (x) and denote by L the fiber P −1 (y).
There exists a differential D x P : T x M → T y Y , which is itself a submetry.The tangent space T x L is the preimage D x P −1 (0) and it is a convex cone in T x M [KL22, Proposition 3.3, Corollary 3.4].We call T x L the vertical space at x and denote it by V x .
The horizontal space H x is the dual cone of T x L in T x M. The cone H x consists of all h ∈ T x M such that ||h|| = |D x P (h)|, where | • | on the right side denotes the distance to the origin of T y Y .
A Lipschitz curve γ : I → M is horizontal if and only if the vector γ ′ (t) is horizontal, for almost all t ∈ I.
For any sequence z j → x in M, any Gromov-Hausdorff limit of (any subsequence of) the vertical spaces V z j contains V x , [KL22, Corollary 8.4].Thus, we find some r 1 > 0 such that B r 1 ∩ L is a C 1,1 -submanifold and such that the following holds true: For any z ∈ B r 1 (x) and any unit vector v ∈ V x , there exists some We call r 1 as above the vertical semicontinuity radius of P at x.
Let r 1 > 0 be given.Let M be a manifold with geometry bounded at x by 1 r 1 .Let P : M → Y be a local submetry, y = P (x) and L = P −1 (y).Let the vertical semicontinuity radius of P at x and the injectivity radius of Y at y be at least r 1 .
Then, upon rescaling M and Y by the constant λ r 1 , we have • For all x 1 , x 2 in the C 1,1 -manifold B 10 (x) ∩ L we have • For any z ∈ B 10 (x) and any unit Proof.We may assume that the constant c appearing in Lemma 2.4 is at least 1.We set λ := 10•c ε and rescale M and Y with λ r 1 .Upon this rescaling, the geometry of M is bounded at x by 1 λ < ε.Hence, the rescaled M is almost flat at x.
The injectivity radius of the rescaled Y at y is at least λ.Therefore, the closest point projection onto L (in the rescaled M) is uniquely defined in B λ (x).Applying Lemma 2.4 we deduce (4.2) and (4.3).
The last point (4.4) follows from the definition of the vertical semicontinuity radius and λ > 10.
Finally, the statement that for some r 0 and all r < r 0 the balls Br (y) are strictly convex is exactly [KL22, Theorem 9.2].Moreover, the proof actually shows that in the present situation one can take r 0 = 10.For some choice of a neighborhood U of x in M, the restriction P : U ∩ S → Y l is a fiber bundle [KL22, Proposition 11.3].Thus, U ∩ S and L ′ := U ∩ P −1 (y ′ ) for any y ′ ∈ Y l are topological manifolds.Since S and L ′ are subsets of positive reach [KL22, Theorem 1.1, Theorem 1.7], both subsets U ∩ S and U ∩ L ′ are C 1,1 -submanifolds of M.
The submanifold S ∩ U in its intrinsic metric is a manifold with locally bounded curvature [KL21, Proposition 1.7].The restriction P : S ∩ U → Y l is a local submetry with all fibers being regular.Thus, this restriction is a In particular the distribution z → V z is continuous on U ∩ S. Thus, the semicontinuity of vertical spaces in M around x implies the following.For a sufficiently small 2δ > 0, any point x 1 ∈ B 2δ (x) ∩ S, any z 1 ∈ B 2δ (x) and any unit vector v ∈ V x 1 there exists some v ′ ∈ V z 1 such that (4.1) holds true.
Thus, δ is the required uniform bound on the vertical semicontinuity radii in a neighborhood of x in S.This finishes the proof.
Lemma 5.1.Let P : M → Y be a local submetry, let L be a fiber of P .Assume x 0 ∈ L is a P -almost flat point.Then Π L : Consider the ball B = B 3 (0) in the horizontal space H p and the subset By the open map theorem [Lyt05b, Lemma 4.1], it suffices to prove (5.1) and that the absolut gradient of the restriction −f : Consider h = exp −1 q (q ′ ) ∈ H q .By (4.3), we find some h Then exp p ( h) ∈ K and from Lemma 2.1 we deduce This proves (5.1).
In order to prove (5.2), we fix a point Denote by w ∈ T q ′′ M the starting direction of the geodesic q ′′ p.
If there exists a vertical unit vector v ∈ V q ′′ = T q ′′ (L ′ ) which encloses an angle less than arccos( 4 5 ) with w, then the first variation formula would imply (5.2).
Assume on the contrary, that such a vertical vector v ∈ V q ′′ does not exist.Then there exists a unit horizontal vector u ∈ H q ′′ which encloses with w an angle at most Since x 0 is a P -almost flat point, we apply (4.4) and (4.3) and find a unit horizontal vector u ′ ∈ H p with |u ′ − u| ≤ 2ε.Then, the angle between the parallel translate ŵ of w to p and û of u ′ to p is at most Note, that ŵ is just the starting direction at p of the geodesic pq ′′ .
Consider the vector ĥ = exp −1 p (p) ∈ H p and the curve η(t) := exp p ( ĥ + t • u ′ ) contained in K and starting at p.By Corollary 2.2, the curve η encloses with vector û an angle less than ε.
Thus, the angle between η and ŵ at p is less than 1 + 4ε < π 2 .Now the first formula of variation implies that d(q ′′ , η(t)) < d(q ′′ , p) for all sufficiently small t.
This contradicts the choice of p.The contradiction finishes the proof of (5.2) and of the Lemma.
Using a combination of Lemma 5.1 and Lemma 2.3 we now provide: Theorem 5.2.P : M → Y a local submetry.Let x 0 ∈ M be such that a neighborhood of x 0 in L = P −1 (P (x 0 )) is a C 1,1 -submanifold.Then there exist r 0 > 0, C > 0 with the following properties, for any 0 < r < r 0 and any z ∈ B r (x 0 ).
(1) For any v ∈ V x 0 there exists (2) For any h ∈ H z there exists (3) For L z = P −1 (P (z)), the closest-point projection Π L : L z ∩ B r (x 0 ) → L is (1 + Cr)-Lipschitz and locally (1 + Cr)-open.The numbers r 0 , C depend only on a bound of the geometry of M at x 0 , a bound on the injectivity radius of Y at P (x 0 ) and the vertical semicontinuity radius of P at x 0 .
Proof.After rescaling we may assume that x 0 is a P -almost flat point.Due to Lemma 4.1, the rescaling constant depend only on a bound of the geometry of M at x 0 , a bound on the injectivity radius of Y at P (x 0 ) and the vertical semicontinuity radius of P at x 0 .
We are going to prove (3) first.By the definition of P -almost flat points, the projection Π L is (1+ε•r)-Lipschitz on the whole ball B r (x 0 ), for any r < 10.Thus, also the restriction of Π L to L ′ ∩ B r (x 0 ) has the same Lipschitz constant.It suffices to improve the openness constant of Π L on L ′ provided by Lemma 5.1.Set r 0 := 1 5 .Let r ≤ r 0 and z ∈ B r (x 0 ) be arbitrary.Set x = Π L (z) and let p be a point on L with a 0 := d(x, p) < ε • r.
We are going to find a point q ∈ L z satisfying Π L (q) = p and (5.3) Extend the geodesic xz to a point z with d(x, z) = 1.Lemma 5.1 provides a point q on the fiber L through z such that d(q, z) ≤ 2a 0 and Π L (q) = p.The geodesics γ := xz and η := pq are horizontal and Consider the point q on the η with d(p, q) = d(x, z).Then P (q) = P (z), hence q ∈ L z .From Lemma 2.3 we deduce (5.3), finishing the proof of (3).
In order to prove (1), we fix r ≤ 1 5 and z ∈ B r (x 0 ).Set again x = Π L (z).Due to (2.9), we have Consider an arbitrary unit vector v ∈ V x 0 .We find a unit vector v ∈ V x with |v − v| ≤ 3εr.
For any sufficiently small δ > 0 consider a point p δ ∈ L with d(x, p δ ) = δ, such that the geodesic xp δ starts in a direction v δ ∈ T x M with |v δ − v| ≤ εr.
Extend as above the geodesic xz until a point z with d(x, z) = 1.Due to Lemma 5.1 we find a point qδ in the fiber L z of z with d(q, z) ≤ 2δ.Let q δ be the point on the geodesic p δ qδ with d(p δ , q δ ) = d(x, z).
Then q δ ∈ L z .From Lemma 2.3, we deduce that the starting direction v δ of the geodesic zq δ satisfies The directions v δ subconverge to a vertical direction v ′ ∈ V z such that |v ′ − v| ≤ 6r.This proves (1).
We now easily deduce: Proof of Proposition 1.3.We cover the compact manifold fiber L by finitely many balls as provided by Theorem 5.2.Choosing a tubular neighborhood B r 0 (L) of L contained in the union of these balls, we obtain the conclusion directly from Theorem 5.2(3).
6. Exponential map in the base 6.1.Exponential map in the base.The following result is a localization of Theorem 1.1.
Proposition 6.1.There exists some µ > 1 with the following properties.Let P : M → Y, x ∈ M, y = P (x) and r 1 be as in Lemma 4.1.
on the ball B 1 (y) ⊂ Y l is 1-Lipschitz.Due to Theorem 6.1 the map f is bilipschitz.By construction, f sends geodesics starting at y to geodesics starting at ȳ. Hence f sends spheres around y onto spheres around ȳ of the same radius.
The restriction of exp ȳ to the concentric sphere ∂B s (0) in T ȳ S n is (1− 1 10 s 2 )-Lipschitz, if we equipp this sphere with its intrinsic metric.Thus, the restriction f : ∂B s (y) → ∂B s (ȳ) is 1-Lipschitz, if both spheres are equipped with their intrinsic metrics.
The bilipschitz map f is differentiable almost everywhere with linear differential, by Rademacher's theorem.By above, at any point z at which f is differentiable, the differential D z f is 1-Lipschitz.We claim that this is enough to conclude that f is 1-Lipschitz.
Indeed, the ball B 1 (y 0 ) can be considered as a Euclidean subset O ⊂ R l with a Lipschitz continuous Riemannian metric.For any vector v in R l , Fubini's theorem implies that for almost every segment γ in O in direction of v, the length of γ in Y is not less than the length of f • γ in S n .On segments parallel to v in O ⊂ Y , the length functional is continuous with respect to uniform convergence.On the other hand, the length of the images f • γ is (as always) lower semi-continuous.Thus, by a limiting procedure, the length of f • γ is not larger than the length of γ for every segment γ in the direction of v. Therefore, the map f is 1-Lipschitz and B 1 (y 0 ) has curvature at most 1.
Another consequence of Theorem 6.1 is the following: Corollary 6.3.Let P : M → Y be a surjective local submetry as above.Let y ∈ Y be an arbitrary point, let r be smaller than the injectivity radius of y and let v ∈ T y Y be a vector with |v| < r.Then the tangent cones T v (T y Y ) and T exp y (v) Y are isometric.
In particular, if exp y (v) is contained in the l-dimensional stratum Y l then v is contained in the l-dimensional stratum (T y Y ) l .
Proof.Consider the geodesic γ v : [0, r) → Y in the direction of v parametrized by arclength.For t ∈ (0, r), the tangent spaces at γ v (t) do not depend on t, [Pet98].Moreover, the tangent space T v (T y Y ) in the Euclidean cone T y Y is isometric T s•v (T y Y ), for all s > 0.
Due to Proposition 6.1, for small s > 0, a neighborhood of (s • v) in T y Y is (1 + Cs 2 )-bilipschitz to a neighborhood of exp y (s • v) in Y , for some C independent of s.Rescaling, letting s go to 0 and using that the tangent cones at s • v respectively at exp y (s • v) do not depend on s, we deduce the claim.This proves the claim and implies that the spaces T γ′ (t) (T γ(t) )Y are pairwise isometric.
Let now l denote the dimension of the maximal Euclidean factor of the pairwise isometric spaces T γ′ (t) (T γ(t) Y ).Then, for all t = s 0 , s 1 , ..., s k as above, the iterated tangent cone T γ′ (t) (T γ(t) Y ) is isometric to T γ(t) Y .By definition, γ(t) is contained in Y l , for all such t.Set S = P −1 (Y l ).Let t ∈ [a, b] be such that γ(t) ∈ Y l .Set x = γ(t) ∈ L t ⊂ S. Then S is a C 1,1 -submanifold of M and the restriction P : S → Y l is a C 1,1 Riemannian submersion.Therefore, the normal vector γ ′ (t) ∈ T x S to L t extends to a unique locally Lipschitz continuous normal field z → ν z ∈ T z S along L t , such that D z P (ν z ) = D x P (ν x ) = D x P (γ ′ (t)) .
Denote by Q t the set of all z ∈ L t such that the geodesic γ z : [a, b] → M with (γ z ) ′ (t) = ν z is defined.Then Q t is an open subset of L t and, if M is complete, Q t = L t .Due to Lemma 7.1, for all z ∈ Q t P • γ z = P • γ = γ .
For all s ∈ [a, b] we obtain a map Hol γ t,s : Q t → L s , the holonomy along γ, given as Hol γ t,s (z) := γ z (s) .Since ν and the exponential map on M are locally Lipschitz, the map Hol γ t,s is locally Lipschitz.If γ(s) ∈ Y l , then Hol γ t,s (Q t ) = Q s and Hol γ t,s and Hol γ s,t are inverse to each other.Thus, Hol γ t,s is locally bilipschitz in this case.Let now r ∈ [a, b] be arbitrary.Find some s ∈ [a, b] such that γ(s) ∈ Y l and |s − r| is smaller than the injectivity radius at γ(r).Then Hol γ s,r • Hol γ t,s = Hol γ t,r .The map Hol γ t,s is locally bilipschitz, as we have seen above.And the map Hol γ s,r : Q s → L r is the closest-point projection to L r .Once s has been chosen close enough to r, we can apply Theorem 5.2 and deduce that the map Hol γ s,r : Q s → L r is locally Lipschitz open.Alltogether we have verified the following

2. 3 .
Comparison of tangent vectors at different points.Let M be a Riemannian manifold with locally bounded curvature.Let O ⊂ M be open, uniquely geodesic and convex.Given x, z ∈ O and vectors v ∈ T x O, w ∈ T z O, we define |v − w| to be the distance in T x O between v and the parallel transport w ′ of w to T x O along the geodesic xz.
Proposition 1.4].On the other hand, any set L of positive reach contains a subset L ′ open and dense in L, which is a C 1,1 -submanifold, possibly with components of different dimensions [RZ17, Theorem 7.5].

Proof.
Upon rescaling, it suffices to prove the statement just for ρ = 1.The distance function d L to L is semiconvex on B 10 (p), see [KL21, Proposition 1.1, Theorem 1.8].Moreover, as shown in [Kle81], see also the proof of [KL21, Proposition 1.1, Proposition 1.3], the semiconvexity constant depends only on the curvature bound and the reach.Thus, there is a universal constant C such that d L is C-semiconvex on B 9 (0).The gradient flow (x, t) → Φ(x, t) of −d L retracts B 4 (p) along the shortest geodesics to L. The C-semiconcavity of −d L implies that the map x → Φ(x, t) is e C•t -Lipschitz continuous [Pet07, Lemma 2.1.4].
Fibers.Let P : M → Y be a surjective local submetry.Let y ∈ Y l ⊂ Y .Then the fiber L = P −1 (y) and the preimage S = P −1 (Y l ) are subsets of positive reach in M [KL22, Theorems 1.1, 1.7].
4. P -almost flatness 4.1.A single point.Let P : M → Y be a local submetry, where M has locally bounded curvature.Fix y ∈ Y and consider L

For a local
submetry P : M → Y we say that x is a P -almost flat point if the conclusions of Lemma 4.1 hold true without rescaling.Due to Lemma 4.1, for any local submetry P : M → Y and any point x ∈ M, such that a neighborhood of x in the fiber L := P −1 (P (x)) is a manifold, M becomes P -almost flat at x upon some rescaling.4.2.Stability along strata.The bound r 1 appearing in Lemma 4.1 can be chosen locally uniformly along strata: Lemma 4.2.Let P : M → Y be a local submetry.Let L be a fiber P −1 (y) and let x ∈ L be a point, such that a neighborhood of x in L is a manifold.Let Y l be the startum through y and S = P −1 (Y ).Then upon rescaling by some µ = µ(M, Y, P, x, y) > 0 the following holds: Any z ∈ B 10 (x) ∩ S is a P -almost flat point.Proof.The curvature and injectivity radii of M are bounded in a fixed ball around x.The injectivity radius of Y is uniformly bounded from below in a neighborhood of y in Y l [KL22, Theorem 11.1].Applying Lemma 4.1 it remains to obtain a uniform lower bound on the vertical semicontinuity radii in a neighborhood of x in S.
is a local submetry if and only if it is locally 1-Lipschitz and locally 1-open.A restriction of a (local) submetry P : X → Y to an open subset O ⊂ X is a local submetry P : O → Y .A local submetry P : X → Y is a global submetry, if X and Y are length spaces and X is proper [KL22, Corollary 2.9].