1 Introduction

We say that a metric space X satisfies the (k, l)-bipolar comparison if for any \(a_0,a_1,\dots ,a_k; b_0,b_1,\dots , b_l \in X\) there are points \({\hat{a}}_0, {\hat{a}}_1,\dots , {\hat{a}}_k, {\hat{b}}_0, {\hat{b}}_1,\dots ,{\hat{b}}_l \) in the Hilbert space \(\mathbb {H}\) such that

$$\begin{aligned} |{\hat{a}}_0-{\hat{b}}_0|_\mathbb {H}=|a_0-b_0|_X,\quad |{\hat{a}}_i-{\hat{a}}_0|_\mathbb {H}=|a_i-a_0|_X,\quad |{\hat{b}}_i-{\hat{b}}_0|_\mathbb {H}=|b_i-b_0|_X \end{aligned}$$

for any i, j and

$$\begin{aligned} |{\hat{x}}-{\hat{y}}|_\mathbb {H}\geqslant |x-y|_X \end{aligned}$$

for any \(x,y\in \{a_0, a_1,\dots , a_k, b_0, b_1,\dots , b_l\}\).

This definition was introduced in [5]. The class of compact length metric spaces satisfying (k, 0)-bipolar comparison with \(k\geqslant 2\) coincide with the class of Alexandrov spaces with nonnegative curvature, (for \(k=2\) it is just one of the equivalent definitions, for arbitrary k see [1, 3]). In general (k, l)-bipolar comparisons (with k or \(l\geqslant 2\)) for length metric spaces are stronger conditions than nonnegative curvature condition and they describe some new interesting classes of spaces. In particular, we prove in [5] that for Riemannian manifolds (4, 1)-bipolar comparison is equivalent to the conditions related to the continuity of optimal transport. Also in [5] we together with coauthors describe classes of Riemannian manifolds satisfying (k, l)–bipolar comparisons for almost all k, l excepting (2, 3) and (3, 3)-bipolar comparisons. In particular it was not known if (3, 3)-bipolar comparison differs from Alexandrov’s comparison. In this note the affirmative answer is obtained as a corollary of some rigidity result for spaces with (3, 3)-bipolar comparison. To formulate exact statements we need some definitions and notations.

Let M be a Riemannian manifold and \(p\in M\). The subset of tangent vectors \(v\in \mathrm {T}_p\) such that there is a minimizing geodesic \([p\,q]\) in the direction of v with length |v| will be denoted as \(\overline{\mathrm{TIL}}_p\). The interior of \(\overline{\mathrm{TIL}}_p\) is denoted by \(\mathrm{TIL}_p\); it is called tangent injectivity locus at p. If at \(\mathrm{TIL}_p\) is convex for any \(p\in M\), then M is called CTIL.

Riemannian manifold M satisfies MTW if the following holds. For any point \(p\in M\), any \(W\in \mathrm{TIL}_p\) and tangent vectors \(X,Y\in \mathrm {T}_p\), such that \(X\perp Y\) we have

$$\begin{aligned} \frac{\partial ^4}{\partial ^2s\,\partial ^2t}\left| \exp _p(s\cdot X)-\exp _p(W+t\cdot Y)\right| _M^2\leqslant 0 \end{aligned}$$
(1)

at \(t=s=0\).

This definition was introduced by Xi-Nan Ma, Neil Trudinger and Xu-Jia Wang in [7], Cedric Villani studied a synthetic version of this definition ( [9]). If the same inequality holds without the assumption \(X\perp Y\) Riemannian manifold M satisfies MTW\(^{\not \perp }\) [2].

MTW and CTIL are necessary condition for TCP (transport continuity property). In [2], Alessio Figalli, Ludovic Rifford and Cédric Villani showed that a strict version of CTIL and MTW provide a sufficient condition for TCP. A compact Riemannian manifold M is called TCP if for any two regular measures with density functions bounded away from zero and infinity the generalized solution of Monge–Ampère equation provided by optimal transport is a genuine (continuous) solution.

Let us denote by \({\mathcal {M}}_{(k,l)}\) the class of smooth complete Riemannian manifolds satisfying (k, l)–bipolar comparison and by \({\mathcal {M}}_{\geqslant 0}\) the class of complete Riemannian manifolds with nonnegative sectional curvature.

It was mentioned above, that

$$\begin{aligned} {\mathcal {M}}_{\geqslant 0}={\mathcal {M}}_{(k,0)} \end{aligned}$$

for \(k\geqslant 2\) and it is obvious from definition, that

$$\begin{aligned} {\mathcal {M}}_{(k',l')}\subset {\mathcal {M}}_{(k,l)} \end{aligned}$$

if \(k'\geqslant k\) and \(l'\geqslant l\). It is proven in [5] that

$$\begin{aligned} {\mathcal {M}}_{\geqslant 0}={\mathcal {M}}_{(2,2)}={\mathcal {M}}_{(3,1)} \end{aligned}$$

and

$$\begin{aligned} {\mathcal {M}}_{(4,1)}= {\mathcal {M}}_{(k,l)} \end{aligned}$$

for \(k\geqslant 4\) and \(l\geqslant 1\). The most interesting fact proven in [5] is that

$$\begin{aligned} {\mathcal {M}}_{(4,1)}= {\mathcal {M}}_{CTIL} \cap {\mathcal {M}}_{MTW^{\not \perp }}, \end{aligned}$$

where \({\mathcal {M}}_{CTIL}\), \( {\mathcal {M}}_{MTW^{\not \perp }}\) are classes of smooth Riemannian manifolds satisfying CTIL and MTW\(^{\not \perp }\) correspondingly. In particular this implies that \({\mathcal {M}}_{(4,1)}\ne {\mathcal {M}}_{\geqslant 0}\).

In this paper we prove the following two results.

Theorem 1.1

Let M be a complete Riemannian manifold that satisfies (3,3)-bipolar comparison and contains a nonempty open flat subset. Then M is flat.

Theorem 1.2

Let M be a complete Riemannian manifold that satisfies MTW\(^{\not \perp }\) and contains a nonempty open flat subset. Then M is flat.

Corollary 1.3

We have that \({\mathcal {M}}_{(3,3)}\ne {\mathcal {M}}_{\geqslant 0}\).

Theorem 1.1 follows from Proposition 2.2 and Theorem 1.2 follows from Proposition 2.3, proved in the next section.

As a related result we would like to mention a rigidity result for manifolds with nonnegative sectional curvature with flat open subsets by Dmitri Panov and Anton Petrunin [8].

2 Proofs

For points a, b, c in a manifold we denote by \(\measuredangle [a\,{}^{b}_{c}]\) the angle at a of the triangle [abc].

Key lemma 2.1

Let M be a complete Riemannian manifold that satisfies (3,3)-bipolar comparison. Assume that for the points \(x_p,p,q,x_q\) in M there is a triangle \([\tilde{p}\tilde{q}\tilde{x}]\) in the Euclidean plane \(\mathbb {E}^2\) such that

$$\begin{aligned} |x_p-p|_M=|\tilde{x}-\tilde{p}|_{\mathbb {E}^2},\quad |p-q|_M=|\tilde{p}-\tilde{q}|_{\mathbb {E}^2},\quad |q-x_q|_M=|\tilde{q}-\tilde{x}|_{\mathbb {E}^2} \end{aligned}$$

and moreover a neighborhood \(N\subset \mathbb {E}^2\) of the base \([\tilde{p}\tilde{q}]\) admits a globally isometric embedding \(\iota \) into M such that \(\iota ([\tilde{p}\tilde{x}]\cap N)\subset [px_p]\) and \(\iota ([\tilde{q}\tilde{x}]\cap N)\subset [qx_q]\). Then \(x_p=x_q\) and the triangle \([pqx_p]\) can be filled by a flat geodesic triangle.

figure a

Proof

Set \(p_-=p\) and \(q_-=q\).

Choose points \(p_0,p_+\in [p_-,x_p]\cap \iota (N)\) so that the points \(p_-,p_0,p_+,x_p\) appear in the same order on \([p_-,x_p]\). Analogously, choose points \(q_0,q_+\in [q_-,x_q]\cap N\) so that the points \(q_-,q_0,q_+,x_p\) appear in the same order on \([q_-,x_q]\). Denote by \(\tilde{p}_-,\tilde{p}_0,\tilde{p}_+,\tilde{q}_-,\tilde{q}_0,\tilde{q}_+\) the corresponding points on the sides of triangle \([\tilde{p}\tilde{q}\tilde{x}]\); so \(\tilde{p}_-=\tilde{p}\) and \(\tilde{q}_-=\tilde{q}\).

Applying the comparison to \(a_0=p_0, a_1=p_-, a_2=p_+, a_3=x_p; \quad b_0=q_0, b_1=q_-, b_2=q_+, b_3=x_q\), we get a model configuration \({\hat{p}}_0, {\hat{p}}_-,{\hat{p}}_+,{\hat{x}}_p,{\hat{q}}_0,{\hat{q}}_-,{\hat{q}}_+,{\hat{x}}_q\) in the Hilbert space \(\mathbb {H}\).

Note that from the comparison it follows that the quadruple \({\hat{p}}_-,{\hat{p}}_0,{\hat{p}}_+,{\hat{x}}_p\) lies on one line and the same holds for the quadruple \({\hat{q}}_-,{\hat{q}}_0,{\hat{q}}_+,{\hat{x}}_q\).

Since

$$\begin{aligned}&|{\hat{p}}_0-{\hat{q}}_+|_{\mathbb {H}}\geqslant |p_0-q_+|_M= |\tilde{p}_0-\tilde{q}_+|_{\mathbb {E}^2},\quad |{\hat{p}}_0-{\hat{q}}_0|_{\mathbb {H}}= |p_0-q_0|_M= |\tilde{p}_0-\tilde{q}_0|_{\mathbb {E}^2},\\&\quad |{\hat{q}}_0-{\hat{q}}_+|_{\mathbb {H}}= |q_0-q_+|_M= |\tilde{q}_0-\tilde{q}_+|_{\mathbb {E}^2}, \end{aligned}$$

we have \(\measuredangle [{\hat{q}}_0\,{}^{{\hat{p}}_0}_{{\hat{q}}_+}]\geqslant \measuredangle [\tilde{q}_0\,{}^{\tilde{p}_0}_{\tilde{q}_+}]\). The same way we get that \(\measuredangle [{\hat{q}}_0\,{}^{{\hat{p}}_0}_{{\hat{q}}_-}]\geqslant \measuredangle [\tilde{q}_0\,{}^{\tilde{p}_0}_{\tilde{q}_-}]\). Since the sum of adjacent angles is \(\pi \), these two inequalities imply that

$$\begin{aligned} \measuredangle [{\hat{q}}_0\,{}^{{\hat{p}}_0}_{{\hat{q}}_\pm }]= \measuredangle [\tilde{q}_0\,{}^{\tilde{p}_0}_{\tilde{q}_\pm }]. \end{aligned}$$

The same way we get that

$$\begin{aligned} \measuredangle [{\hat{p}}_0\,{}^{{\hat{q}}_0}_{{\hat{p}}_\pm }]= \measuredangle [\tilde{p}_0\,{}^{\tilde{q}_0}_{\tilde{p}_\pm }]. \end{aligned}$$

From the angle equalities, we get that

$$\begin{aligned} |{\hat{p}}_--{\hat{q}}_+|_{\mathbb {H}}\leqslant |\tilde{p}_--\tilde{q}_+|_M \end{aligned}$$
(1)

and the equality holds if the points \({\hat{p}}_-, {\hat{q}}_+\) lie in one plane and on the opposite sides from the line \({\hat{p}}_0 {\hat{q}}_0\). By (3,3)-bipolar comparison the equality in 1 indeed holds.

It follows that configuration \({\hat{p}}_0, {\hat{p}}_-,{\hat{p}}_+,{\hat{x}}_p,{\hat{q}}_0,{\hat{q}}_-,{\hat{q}}_+,{\hat{x}}_q\) is isometric to the configuration \(\tilde{p}_0, \tilde{p}_-,\tilde{p}_+,\tilde{x},\tilde{q}_0,\tilde{q}_-,\tilde{q}_+,\tilde{x}\); in particular, \({\hat{x}}_q={\hat{x}}_p\).

By (3,3)-bipolar comparison \(|x_p-x_q|_M\leqslant |{\hat{x}}_q-{\hat{x}}_p|_{\mathbb {H}}\); therefore \(x_p=x_q\); so we can set further \(x=x_p=x_q\).

Note that we also proved that the angles at p and q in the triangle [pqx] coincide with their model angles; that is,

$$\begin{aligned} \measuredangle [p\,{}^q_x]=\measuredangle [{\tilde{p}}\,{}^{\tilde{q}}_{\tilde{x}}], \quad \measuredangle [q\,{}^p_x]=\measuredangle [{\tilde{q}}\,{}^{\tilde{p}}_{\tilde{x}}]. \end{aligned}$$

By the lemma on flat slices (see for example Lemma 2.1 in [4]), there is a global isometric embedding \(\iota '\) of the solid model triangle \([\tilde{p}\tilde{q}\tilde{x}]\) to M which sends \([\tilde{p}\tilde{q}]\) to [pq] and \([\tilde{p}\tilde{x}]\) to [px]. Note that \(\iota '\) has to coincide with \(\iota \) on N. It follows that \(\iota '\) maps \([\tilde{q}\tilde{x}]\) to [qx], which finishes the proof. \(\square \)

Theorems 1.1 and 1.2 follow from the propositions below.

Proposition 2.2

Let M be a complete Riemannian manifold that satisfies (3,3)–bipolar comparison. Then any point \(x\in M\) admits a neighborhood \(U\ni x\) such that if U contains a nonempty open flat subset, then U is flat.

Proof

Given a point p consider a convex neighborhood \(U\ni p\) such that injectivity radius at any point of U exceeds the diameter of U; in particular any two points \(p,q\in U\) are connected by unique minimizing geodesic [pq] which lies in U. Denote by F an open flat subset in U; we can assume that F is convex. \(\square \)

Note that by the key lemma we have the following:

Claim

For any \(x\in U\) and any \(p,q\in F\) the triangle [pqx] admits a geodesic isometric filling by a flat triangle.

Indeed, set \(x_p=x\). Consider a plane triangle \([\tilde{p}\tilde{q}\tilde{x}]\) that has the same angle at \(\tilde{p}\) and the same adjacent sides as the triangle [pqx]. Since F is flat and convex there is a flat open geodesic surface \(\Sigma \) containing [pq] and a part of [px] near p. Choose a direction at q that runs in \(\Sigma \) at the angle \(\measuredangle [{\tilde{q}}\,{}^{\tilde{p}}_{\tilde{x}}]\) to [qp]. Consider the geodesic in this direction of the length \(|\tilde{q}\tilde{x}|\). Since diameter of U exceeds the injectivity radius at q, this geodesic is minimizing. It remains to apply the key lemma.

From the claim, it follows that the sectional curvature \(\sigma _x(X,Y)\) vanishes for any point \(x\in U\) and any two velocity vectors \(X,Y\in \mathrm {T}_x\) of minimizing geodesics from x to F. Since the set of such sectional directions is open, curvature vanish at x; hence the result.

Proposition 2.3

Let M be a complete Riemannian manifold that satisfies MTW\(^{\not \perp }\). Then any point \(p\in M\) admits a neighborhood \(U\ni p\) such that if U contains a nonempty open flat subset, then U is flat.

Proof

For a given \(p\in M\) let us take a neighborhood \(U\ni p\) as in the proof of the previous proposition. The same proof as (Thm 1.2 [5] ) shows that U satisfies (4, 1)-bipolar comparison (CTIL condition is not necessary, because we stay away from the cut-locus). Again, same proof as (the Thm 1.2 [5]) shows that inside this neighborhood (4, 1)-bipolar comparison is equivalent to (4, 4)-bipolar comparison. Further note that (4, 4)-bipolar comparison implies (3, 3)-bipolar comparison. Now we can follow the same lines as in the proof of Proposition 2.2, because (3, 3)-bipolar comparison is used only locally in the proof. \(\square \)