About every convex set in any generic Riemannian manifold

Alexander Lytchak; Anton Petrunin

doi:10.1515/crelle-2021-0058

Published by De Gruyter October 26, 2021

About every convex set in any generic Riemannian manifold

Alexander Lytchak and Anton Petrunin

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2021-0058

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Abstract

We give a necessary condition on a geodesic in a Riemannian manifold that can run in some convex hypersurface. As a corollary, we obtain peculiar properties that hold true for every convex set in any generic Riemannian manifold (M,g). For example, if a convex set in (M,g) is bounded by a smooth hypersurface, then it is strictly convex.

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: 281071066

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-2005279

Funding statement: Alexander Lytchak was partially supported by the DFG grant No. 281071066, TRR 191. Anton Petrunin was partially supported by the NSF grant No. DMS-2005279.

A Normalization of metrics

This appendix is devoted to the algebra of curvature tensor and its covariant derivatives that leads to a proof of Claim 3.1.

Choose an m-dimensional Euclidean space T. Denote by 𝒮 the space of self-adjoint operators on T.

Consider the space 𝒢 of germs of Riemannian metrics on T at 0 that coincide with the canonical metric at 0. Any germ in 𝒢 can be described by 〈G⋅𝗏,𝗐〉, where 𝗑↦G𝗑 is a smooth function T→𝒮 such that G0=id.

The k-jet of G is defined by the Taylor polynomial of G of degree k

(A.1)G𝗑=id+G𝗑1+…+G𝗑k+o⁢(|𝗑|k),

where 𝗑↦G𝗑i is a homogeneous polynomial T→𝒮 of degree i.

We note that every array of homogeneous polynomials G1,…,Gk:T→𝒮 such that deg⁡Gi=i appears in (A.1) for the germ in 𝒢 defined by

(A.2)G𝗑=id+G𝗑1+…+G𝗑k.

The space of k-jets of germs in 𝒢 will be denoted by 𝒢k.

A germ in 𝒢 will be called normal if the standard coordinates on T coincide with normal coordinates of the germ in a neighborhood of the origin. By the Gauss lemma, a germ defined by G is normal if and only if

(A.3)G𝗑⋅𝗑=𝗑

for all small 𝗑∈T. The subspace of normal germs in 𝒢 and their k-jets will be denoted by 𝒩 and 𝒩k, respectively.

Suppose that G describes a germ in 𝒩 and G1,…,Gk be as in (A.1). By (A.3)

(A.4)G𝗑i⋅𝗑=0

for any i. Moreover, for an array of polynomials G1,…,Gk:T→𝒮 such that Gi is homogeneous of degree i and (A.4) holds for each i, the sum (A.2) defines a normal k-jet; that is, (A.4) is the only condition on the normality of jets.

Christoffel symbols vanish in normal coordinates, thus, G1=0, for G∈𝒩.

Choose 𝗑∈T; denote by 𝒮𝗑 the subspace of the operators S∈𝒮 such that S⋅𝗑=0. By (A.4), G𝗑i∈𝒮𝗑 for any germ in 𝒩. The following claim says that G𝗑i can be chosen arbitrarily in 𝒮𝗑 for i≥2 and 𝗑≠0.

Claim A.1.

Given x≠0 in T and a sequence of operators A2,…,Ak∈Sx, there is a germ (G1,…,Gk) in Nk such that Gxi=Ai for any i≥2.

Proof.

For any unit vector 𝗒 in T perpendicular to 𝗑, consider the orthogonal projection P𝗒 in T onto the line generated by 𝗒. Diagonalizing operators in 𝒮𝗑, we see that such projections P𝗒 generate 𝒮𝗑 as a vector space.

The subspace 𝒩k is described by (A.4), hence it defines a linear subspace of 𝒮𝗑k. Thus, it suffices to verify the following: For any 2≤j≤k and any unit vector 𝗒 in T perpendicular to 𝗑, there exists a germ (G1,…,Gk) in 𝒩k such that G𝗑j=P𝗒 and Gi=0 for i≠j.

Such a normal germ can be constructed as a product of a surface of revolution (corresponding to the (𝗑,𝗒)-plane) and a Euclidean space. ∎

Suppose that a germ in 𝒢 is described by G:T→𝒮. Consider its array of Jacobi operators (R1,…,Rk) at the origin; recall that R1=0. The identities in Section 2 imply that any such array (R1,…,Rk) belongs to the space ℛk defined by the following conditions:

each Ri:T→𝒮 is a homogeneous polynomial,
deg⁡Ri=i,
R𝗑i⋅𝗑=0 for any i and 𝗑∈T.

Note that these conditions are exactly the same as for Gi in 𝒩k. Therefore, ℛk can be identified with 𝒩k, but we will keep separate notations for them.

The expression of the curvature tensor in terms of the metric and its derivatives defines a natural algebraic map

ρk:𝒢k→ℛk.

For any k≥2, any G∈𝒩k and (R1,…,Rk)=ρk⁢(G), we have

Gk=ak⋅Rk+Ak,

where ak is a nonzero constant and Ak is a field of self-adjoint operators that can be written as a polynomial of R2,…,Rk-2. This statement follows easily from the formula derived by Oldřich Kowalski and Martin Belger [9, Proposition 2.2]. (In fact, ak=-2⋅k-1k+1, but we will not need it.)

Hence, the map ρk admits an algebraic inverse map::

Claim A.2.

The restriction ρk|Nk is an algebraic diffeomorphism Nk↔Rk.

Applying Claim A.1, we get the following:

Corollary A.3.

Given x≠0 in T and a sequence of operators A2,…,Ak∈Sx, there is a germ Nk with Jacobi operators Rxi=Ai for any i≥2.

Proposition A.4.

The map ρk:Gk→Rk is an algebraic submersion. (See Figure 1.)

Proof.

Evidently, ρk is algebraic.

Figure 1

Any germ in 𝒢 becomes normal if the space T is reparametrized by its exponential map. This defines the normalization map 𝒢→𝜈𝒩. Since the curvature tensors does not change under this (or any other) coordinate change, it follows that ν commutes with ρk:𝒢,𝒩→ℛk.

By Claim A.2, 𝒩k↔ρkℛk is a diffeomorphism. The maps

𝒢k→ρkℛk↔ρk𝒩k

together with the forgetful maps 𝒢→𝒢k and 𝒩→𝒩k commute. In particular, we get a map 𝒢k→νk𝒩k that commutes with the forgetful maps and the normalization ν. Hence, νk and ρk commute.

Note that the inclusion 𝒩⁢↪𝜄⁢𝒢 is a right inverse of ν. Moreover, by changing the parametrization on T to normal coordinates of a given germ G in 𝒢, we may assume that G lies in the image of ι. Therefore, there is an inclusion 𝒩k⁢↪ιk⁢𝒢k that is a right inverse of ν such that its image contains any given jet in 𝒢k. It follows that 𝒢k→νk𝒩k is a submersion, hence the result. ∎

Proof of Claim 3.1.

Denote by 𝒢~k the space of all k-jets of Riemannian metrics at a given point p. Denote by Σ~k all jets in 𝒢~k such that for some nonzero tangent vector 𝗑∈Tp the Jacobi operators R𝗑2,…,R𝗑k have a common exceptional invariant subspace.

By Tarski–Seidenberg theorem, Σ~k is semialgebraic; in particular it is stratified. Due to the Thom transversality theorem [7, Theorem 2.3.2], it is sufficient to show that for any point p the codimension of Σ~k in 𝒢~k is larger than m=dim⁡M.

This is a pointwise statement; therefore we may fix p from now on.

A jet in 𝒢~0 is described by the metric tensor g0 on T=Tp⁢M. Note that the forgetful map 𝒢~k→𝒢~0 is a fiber bundle. Furthermore, the restriction of this forgetful map to Σ~ is also a fiber bundle. Thus, it suffices to prove that the intersection Σk of Σ~k with a fiber of 𝒢~k→𝒢~0 has codimension at least m. Note that the fiber of the forgetful map over the Euclidean structure on T given by g0 is exactly the space 𝒢k investigated above.

In other words, if we choose a chart T→M, then g0 defines an inclusion 𝒢k↪𝒢~k, and it is sufficient to show that

(A.5)codim⁡Σk→∞ as ⁢k→∞;

here we consider Σk=Σ~k∩𝒢k as a subset of 𝒢k.

Denote by ℒ the semialgebraic set of all pairs (L,𝗑), where L is a subspace of T such that 1<dim⁡L<m, and 𝗑∈L∖{0}. Given (L,𝗑)∈ℒ, denote by Σk⁢(L,𝗑) the subset of jets in 𝒢k such that L is an invariant subspace of all Jacobi operators R𝗑i for any i≤k.

Choose (L,𝗑)∈ℒ. We claim that Proposition A.4 implies

(A.6)codim⁡Σk⁢(L,𝗑)→∞ as ⁢k→∞.

Indeed, a normal germ (G1,…,Gk) belongs to Σk⁢(L,𝗑) if and only if all the Jacobi operators R𝗑2,…,R𝗑k∈𝒮𝗑 have invariant subspace L. The codimension of the space of (k-1)-tuples in Sx that all have L as invariant subspace grows with k. By Proposition A.4 and Corollary A.3 the composition 𝒢k→ℛk→𝒮𝗑k-1 that sends a germ to the array of its Jacobi operators (R𝗑2,…,R𝗑k) is a submersion. Therefore, (A.6) follows.

Observe that

codim⁡Σk≥codim⁡Σk⁢(L,𝗑)-dim⁡ℒ.

Therefore, (A.5) follows. ∎

B Final remarks

We expect that the following question admits an affirmative answer.

Question B.1.

Is it true that any Riemannian manifold (M,g) contains a nontrivial geodesic that runs in the boundary of some convex subset?

There is a good chance that the argument of Albert Borbély [3, Lemma 2.1] can be modified to answer the following question. Assuming that the answer is affirmative, it can be combined with the main proposition to derive further restrictions on convex hulls in generic Riemannian manifolds.

Question B.2.

Let ℭ be the closure of a convex hull of a set Q in a Riemannian manifold. Then all points of ℭ with rank at most 1 lie on minimizing geodesics between points in Q.

The presented argument, when properly extended to infinite-dimensional manifolds, might lead to a negative answer to the following question of Mikhael Gromov [8, 6.B(f)1].

Question B.3.

Let X be a complete CAT⁡(0) space (not necessarily locally compact). Is it true that any compact set of X lies in a compact convex subset?

A surprising behavior of convex sets in complete (but not locally compact) CAT⁡(0) spaces is discussed by Nicolas Monod [12].

Finally, let us mention that there is a result of Anatoliy Milka [11, Section 4] about rank of points on geodesics in the intrinsic metric of convex surfaces; it is closely related to our main proposition but goes in the opposite direction.

Acknowledgements

We thank Mohammad Ghomi and Frederick Wilhelm for their interest in our result, the anonymous referee for helpful criticism.

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Received: 2021-04-14

Revised: 2021-08-02

Published Online: 2021-10-26

Published in Print: 2022-01-01

About every convex set in any generic Riemannian manifold

Abstract

A Normalization of metrics

Claim A.1.

Proof.

Claim A.2.

Corollary A.3.

Proposition A.4.

Proof.

Proof of Claim 3.1.

B Final remarks

Question B.1.

Question B.2.

Question B.3.

Acknowledgements

References

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