---
abstract: 'We generalize the Grove-Searle symmetry rank bound to the setting of positive intermediate Ricci curvature using a local argument, only requiring commuting Killing fields around a point. We address ramifications of this result related to the Maximal Symmetry Rank Conjecture for non-negative curvature. We also prove that our symmetry rank bound is locally optimal: Every Riemannian metric is arbitrarily close in the $C^1$-topology to one that achieves the upper bound at a point.'
address: |
    900 University Avenue\
    Skye Hall Room 278\
    Riverside, CA 92521
author:
- Lawrence Mouillé
bibliography:
- 'symmetry\_rank\_biblio.bib'
title: |
    Local symmetry rank bound for\
    positive intermediate Ricci curvatures
---

Introduction
============

The <span>*symmetry rank*</span> of a Riemannian manifold $(M,g)$, denoted $\operatorname{symrank}(M,g)$, is the rank of its isometry group, i.e. the maximal dimension of a torus that can act isometrically and effectively on $(M,g)$. In this article, we establish a symmetry rank bound for manifolds with *positive intermediate Ricci curvature*:

\[thm:global\] If $M$ is a connected Riemannian $n$-manifold with $\operatorname{Ric}_k>0$ for some $k\in\{1,\dots,n-2\}$, then $$\operatorname{symrank}(M,g)\leq\left\lfloor\frac{n+k}{2}\right\rfloor.$$ The same conclusion holds if instead we assume $M$ is closed, connected, and contains a point at which $\operatorname{Ric}_k>0$ for some $k\in\{1,\dots,n-2\}$.

Because $\operatorname{Ric}_1>0$ is equivalent to $\sec>0$, this result generalizes the symmetry rank bound established by Grove and Searle for positively curved manifolds in @maxsymrank:

\[maxsymrank\] Any closed, connected Riemannian $n$-manifold $M$ with positive sectional curvature has $$\operatorname{symrank}(M,g)\leq\left\lfloor\frac{n+1}{2}\right\rfloor,$$ and in the case of equality, $M$ is diffeomorphic to $S^n$, $\mathbb{R}\mathrm{P}^n$, $\mathbb{C}\mathrm{P}^{n/2}$, or a lens space.

Theorem \[thm:global\] is a consequence of Theorem \[thm:local\] below, which we state after a discussion about positive intermediate Ricci curvature.

Positive intermediate Ricci curvature
-------------------------------------

In this section, we define positive intermediate Ricci curvature, discuss some structure results and examples, and present motivation for studying positive intermediate curvature with symmetries.

Let $(M,g)$ be an $n$-dimensional Riemannian manifold, and let $k\in\{1,\dots,n-1\}$. Given a $(k+1)$-dimensional subspace $\mathcal{V}\subseteq T_pM$, let $R^\mathcal{V}$ denote the type-$(1,3)$ curvature tensor $R$ restricted to $\mathcal{V}^3$ composed with orthogonal projection to $\mathcal{V}$. Then given a unit vector $u\in\mathcal{V}$, define the <span>*$k^\mathrm{th}$-intermediate Ricci curvature*</span> for the pair $(u,\mathcal{V})$ to be $$\operatorname{Ric}_k(u,\mathcal{V})=\operatorname{trace}\left(x\mapsto R^\mathcal{V}(x,u)u\right)=\sum_{i=1}^k\sec(u,e_i),$$ where $\{e_1,\dots,e_k\}$ is any orthonormal basis for the orthogonal complement of $\mathrm{span}\{u\}$ in $\mathcal{V}$.

Notice that $\operatorname{Ric}_1(u,\mathcal{V})$ is the sectional curvature of the $2$-plane $\mathcal{V}$, and\
$\operatorname{Ric}_{n-1}(u,T_pM)$ is the Ricci curvature $\operatorname{Ric}(u,u)$.

We say $(M,g)$ has <span>*positive $k^\mathrm{th}$-intermediate Ricci curvature*</span> if\
$\operatorname{Ric}_k(u,\mathcal{V})>0$ for all pairs $(u,\mathcal{V})$ in the tangent bundle $TM$. We abbreviate this by writing $\operatorname{Ric}_k(M,g)>0$, omitting $M$ or $g$ when they are understood.

Thus $\operatorname{Ric}_1>0$ is equivalent to $\sec>0$, and $\operatorname{Ric}_{n-1}>0$ is equivalent to $\operatorname{Ric}>0$. Furthermore, it is easy to check that if $\operatorname{Ric}_k>0$ for some $k\in\{1,\dots,n-1\}$, then $\operatorname{Ric}_l>0$ for all $l\geq k$. In particular, $\operatorname{Ric}_k>0$ implies that $\operatorname{Ric}>0$. So if a compact manifold $M$ has $\operatorname{Ric}_k>0$, then by the Bonnet-Myers theorem, $\pi_1(M)$ is finite. We also have the following structure results for $\operatorname{Ric}_k>0$ that go beyond what has been established for $\operatorname{Ric}>0$:

-   If $M$ is open and $\operatorname{Ric}_k(M)>0$, then $M$ has the homotopy type of a CW-complex with cells each of dimension $\leq k-1$. @shenkth

-   Suppose $\operatorname{Ric}_k(M)\geq k$ and the first systole of $M$ $>\pi\sqrt{\frac{k-1}{k}}$.

    1.  If $M$ is even-dimensional and orientable, then it is simply connected.

    2.  If $M$ is odd-dimensional, then it is orientable. @wilhelminterm

-   If $\operatorname{Ric}_k(M)\geq k$ and $M$ contains a point for which the conjugate radius $>\frac{\pi}{2}$, then the universal cover of $M$ is $(n-k)$-connected. @softconnprinc

For more, see @yousef, @restrictions, @focalradius, @GumaerWilhelm, @KettererMondino [Theorem 6.1], @Shen_sphere, @wilhelminterm [Theorem 11], @connprinc [Remark 2.4], or @wu. It appears there have been no attempts to systematically document examples of manifolds with positive intermediate Ricci curvature which do not have positive sectional curvature. Here, we list two sources of examples:

-   Consider any positively curved manifolds $M^m$ and $N^n$. It is an easy exercise to show that with respect to the product metric, $\operatorname{Ric}_k(M^m\times N^n)>0$ only for $k\geq \max\{m,n\}+1$. Improving upon this, the author has shown that products of positively curved homogeneous spaces admit metrics with $\operatorname{Ric}_2>0$ @newexamples.

-   It follows from @Nash that a compact homogeneous space $G/H$ admits a metric with positive Ricci curvature if and only if $\pi_1(G/H)$ is finite. Furthermore, if the rank of $G/H$ is $r$, then $\operatorname{Ric}_{r-1}(G/H)\not>0$. Thus $\operatorname{Ric}_k(G/H)>0$ for some $k\in\{\operatorname{rank}(G/H),\dots,\dim(G/H)-1\}$.

In the setting of positive curvature, the Grove symmetry program has been effective in inspiring innovative results. The goal of this program is to classify positively curved manifolds that have large isometry groups. This initiative has led to advances in global Riemannian geometry, providing symmetry obstructions for positive curvature like the @maxsymrank. Attempts to uncover connections between curvature and symmetries have also helped in creating new examples of positive or non-negative curvature, a large area of research in its own right. Because the Grove symmetry program has proven to be so fruitful, the author's intention is to initiate a study of interactions between positive intermediate Ricci curvature and symmetries.

Local symmetry rank bound
-------------------------

In this section, we introduce the key results of the article, Theorem \[thm:local\] and Theorem \[thm:dense\].

The symmetry rank bound from the Grove-Searle @maxsymrank is proven using global arguments. It relies on globally defined torus actions and a theorem of Berger stating that any Killing field on an even dimensional positively curved manifold has a zero. Wilking obtained the same bound for quasi-positive curvature using only the Gauss equation @wilkingunpublished, and Galaz-García included Wilking's argument in @symrankquasiposlowdim. Searle and Wilhelm noticed that Wilking's argument only requires commuting Killing fields and positive sectional curvature for planes spanned by the Killing fields @perscomm.

Informed by these observations, our first key result is a local symmetry rank bound for positive intermediate Ricci curvature:

\[thm:local\] Suppose that $M$ is a Riemannian $n$-manifold, $N\subset M$ is a submanifold through a point $p$, the tangent space $T_pN$ is spanned by commuting Killing fields of $N$, and $k\in\{1,\dots,n-2\}$. If $\operatorname{Ric}_k(u,\mathcal{V})>0$ for all $(k+1)$-dimensional subspaces $\mathcal{V}\subseteq T_pN$ and all $u\in\mathcal{V}$, then $$\dim(N)\leq \left\lfloor\frac{n+k}{2}\right\rfloor.$$

Note that the submanifold $N$ in Theorem \[thm:local\] is not assumed to be complete; it is only required to be defined in a neighborhood of the point $p$. Also notice the utility of Theorem \[thm:local\] is that it is local in nature, and thus can be applied in many scenarios. Now when a torus acts isometrically and effectively on a Riemannian manifold, the principal isotropy groups are trivial, and the tangent spaces to any principal orbit are spanned by commuting Killing fields. Thus, the first statement in Theorem \[thm:global\] follows immediately from Theorem \[thm:local\]. The second statement in Theorem \[thm:global\] follows by applying Theorem \[thm:local\] on the principal orbits of a torus action with rank $>\frac{n+k}{2}$ and then using a compactness argument to reach a contradiction.

Now, we establish terminology for when the upper bound in Theorem \[thm:local\] is achieved:

Suppose $M$ has a submanifold $N$ such that for some point $p\in N$, $T_pN$ is spanned by commuting Killing fields of $N$, and $\operatorname{Ric}_k(u,\mathcal{V})>0$ for all $(k+1)$-dimensional subspaces $\mathcal{V}\subseteq T_pN$ and all $u\in\mathcal{V}$. If $\dim(N)=\lfloor\frac{n+k}{2}\rfloor$, then we say $M$ has <span>*$k$-maximal local symmetry rank*</span> at $p$.

Notice that no manifold admits $(n-1)$-maximal local symmetry rank: If $k=n-1$, then $\left\lfloor\frac{n+k}{2}\right\rfloor=n-1<n=k+1$. So when $k=n-1$, there can be no $(k+1)$-dimensional subspaces $\mathcal{V}$ of $T_pN$ while $\dim(N)=\lfloor\frac{n+k}{2}\rfloor$.

Recall from the that the list of manifolds which admit positive sectional curvature and maximal global symmetry rank is restrictive, only consisting of $S^n$, $\mathbb{R}\mathrm{P}^n$, $\mathbb{C}\mathrm{P}^{n/2}$, or lens spaces. In stark contrast, we have the following density result for $k$-maximal local symmetry rank:

\[thm:dense\] Let $M$ be an $n$-manifold, $n\geq 3$, $k\in\{1,\dots,n-2\}$, and $p\in M$. Every Riemannian metric $g$ on $M$ is arbitrarily close in the $C^1$-topology to a metric $\tilde{g}$ such that $(M,\tilde{g})$ has $k$-maximal local symmetry-rank at $p$.

Theorem \[thm:dense\] shows that the upper bound in Theorem \[thm:local\] can be realized on any manifold. We prove Theorem \[thm:dense\] by sewing in a model metric on a small ball around the point $p$ and showing that the resulting metric can be made arbitrarily close in the $C^1$-distance to the original. These model metrics are constructed on $\mathbb{R}^n$ and have $k$-maximal local symmetry rank at the origin.

Global symmetry rank
--------------------

In this section, we discuss the global symmetry rank bound in Theorem \[thm:global\]. Comparing Theorem \[thm:global\] to the leads naturally to the following:

Is the symmetry rank bound in Theorem \[thm:global\] optimal? Can we classify $n$-manifolds with $\operatorname{Ric}_k>0$ that have large symmetry-rank?

In order to make progress towards addressing these and other questions concerning positive intermediate Ricci curvature, it would be beneficial to find non-trivial examples of manifolds with $\operatorname{Ric}_k>0$. By this, we suggest finding and cataloging examples where $\operatorname{Ric}_k>0$ while $\operatorname{Ric}_{k-1}\not>0$ for $k\geq 2$. See @newexamples for new examples of manifolds with $\operatorname{Ric}_2>0$, many of which do not admit metrics with $\sec>0$.

In the $k=n-2$ case, the symmetry rank bound from Theorem \[thm:global\] states that $\operatorname{symrank}(M,g)\leq\big\lfloor\frac{n+(n-2)}{2}\big\rfloor=n-1$. Cohomogeneity-one torus actions actions may occur on Ricci-positive manifolds in dimensions $2$ and $3$; e.g. the $T^1$-action on $S^2$ and the $T^2$-action on $S^3$. However, it is known that in dimensions $\geq 4$, closed manifolds that admit cohomogeneity-one torus actions must have infinite fundamental group @pak [@parker]. Thus by the Bonnet-Myers theorem, such manifolds cannot admit invariant metrics of positive Ricci curvature, and hence cannot admit $\operatorname{Ric}_{n-2}>0$. Therefore, we have the following:

If $M$ is a closed, connected $n$-manifold with $\operatorname{Ric}_k>0$ for $n\geq 4$ and $k\geq n-2$, then $$\operatorname{symrank}(M,g)\leq n-2.$$

Corro and Galaz-García show in @riccisymrank that for each dimension $n \geq 6$, there exist examples of $n$-manifolds which admit a metric of positive Ricci curvature with $\operatorname{symrank}(M,g)=n-4$. It remains to be shown if this can be improved to give examples of positive Ricci curvature with $\operatorname{symrank}(M,g)=n-2$.

Ramifications in non-negative curvature
---------------------------------------

In this section, we discuss a consequence of Theorem \[thm:global\] in the context of non-negative curvature. Specifically, Galaz-García and Searle state the for non-negatively curved manifolds in @maxsymrankconj, which has since been sharpened by Escher and Searle in @eschersearle:

\[maxsymrankconj\] Let $(M,g)$ be a closed, simply connected, $n$-dimensional Riemannian manifold with non-negative sectional curvature. Then

1.  $\displaystyle\operatorname{symrank}(M,g)\leq\left\lfloor\frac{2n}{3}\right\rfloor$, and \[conjbound\]

2.  in the case of equality, $M$ is equivariantly diffeomorphic to a product of spheres or a quotient thereof by a free linear action of a torus of rank less than or equal to $2n \;\mathrm{mod}\; 3$. \[diffeoclass\]

It was shown that the holds for $4\leq n\leq 6$ by Galaz-García and Searle in @maxsymrankconj, it holds for $7\leq n\leq 9$ by Escher and Searle in @eschersearle, and also Part (\[conjbound\]) holds for $10\leq n\leq 12$ in @eschersearle. Furthermore, the has been confirmed in all dimensions for torus actions that are *isotropy-maximal* in @eschersearle. With the assumption of non-negative curvature replaced with *rational ellipticity*, Part (\[conjbound\]) has been established in all dimensions along with a rational homotopy theoretic version of Part (\[diffeoclass\]) by Galaz-García, Kerin, and Radeschi in @toruselliptic. This result is relevant to the conjecture above because the Bott Conjecture claims that any non-negatively curved manifold is rationally elliptic.

Notice that a non-negatively curved manifold can have positive $k^\mathrm{th}$-intermediate Ricci curvature for $k\geq 2$ without being positively curved. If $\operatorname{Ric}_k>0$ at a point for $k\leq\frac{n}{3}$, then by Theorem \[thm:global\], $\operatorname{symrank}(M,g)\leq\lfloor\frac{2n}{3}\rfloor$. In particular, we have the following:

Let $(M,g)$ be a closed, simply connected Riemannian manifold with non-negative section curvature and dimension $n\geq 13$. If $M$ contains a point at which $\operatorname{Ric}_{\lfloor n/3\rfloor}>0$, then the conclusion of Part (\[conjbound\]) of the holds.

Structure of this article
-------------------------

In Section \[sec:proof\], we prove Theorem \[thm:local\]. In Section \[sec:example\], we construct metrics on $\mathbb{R}^n$ that have $k$-maximal local symmetry rank at the origin for $n\geq 3$ and $k\in\{1,\dots, n-2\}$. In Section \[sec:close\], we prove Theorem \[thm:dense\] by establishing a result about sewing metrics in the $C^1$-topology and applying it to the metrics constructed in Section \[sec:example\].

Acknowledgments
---------------

This article is part of my doctoral thesis at the University of California, Riverside, and many of the results were completed while I was supported by the university's Dissertation Year Program Award. I thank my thesis advisor, Fred Wilhelm, for his insights and valuable feedback during the preparation of this article. I also thank Priyanka Rajan, Karsten Grove, and Guofang Wei for helpful discussions concerning this project. Finally, I thank the referee, whose detailed comments and suggestions led to significant improvements of the exposition and presentation in this article.

Symmetry rank bound {#sec:proof}
===================

In this section, we prove Theorem \[thm:local\]. Our argument is informed by Wilking's proof of the symmetry rank bound for quasi-positive curvature @wilkingunpublished, which may be found in Galaz-García's article @symrankquasiposlowdim.

Assume that $M$ and $N$ are as in Theorem \[thm:local\]. Let $\operatorname{I\!I}$ denote the second fundamental form for $N\subset M$, i.e. for $u,v\in T_pN$, $\operatorname{I\!I}(u,v)=(\nabla_uV)^\perp\in T_pN^\perp$, where $V$ is any extension of $v$ to a vector field. Given a unit vector $u\in T_pN$, let $\mathcal{O}_u$ denote the orthogonal complement of $\mathrm{span}\{u\}$ in $T_pN$. Now considering $\operatorname{I\!I}(u,\cdot)$ as a linear map $\mathcal{O}_u\to T_pN^\perp$, we have the following:

\[lem:ker\] If $\dim(N)>\frac{n+k}{2}$, then $\dim(\ker\operatorname{I\!I}(u,\cdot))\geq k$.

If $\dim(N)>\frac{n+k}{2}$, then $$\begin{aligned}
            \dim(\ker \operatorname{I\!I}(u,\cdot))=\dim (\mathcal{O}_u)-\dim({\mathrm{Im}}\operatorname{I\!I}(u,\cdot))&\geq\left(\dim(N)-1\right)-\left(n-\dim(N)\right)\\
            &>\left(\frac{n+k}{2}-1\right)-\left(n-\frac{n+k}{2}\right).
        \end{aligned}$$ Therefore, $\dim(\ker\operatorname{I\!I}(u,\cdot))> k-1$.

We now prove our main theorem using Lemma \[lem:ker\] and the Gauss equation.

\[Proof of Theorem \[thm:local\]\] Suppose $\dim(N)>\frac{n+k}{2}$, and fix a unit vector $u\in T_pN$ such that $|\operatorname{I\!I}(u,u)|\geq0$ is minimal. By Lemma \[lem:ker\], we can choose orthonormal $e_1,\dots,e_k\in \mathcal{O}_u$ such that $\operatorname{I\!I}(u,e_i)=0$. For $i=1,\dots,k$, define $f_i:\mathbb{R}\to\mathbb{R}$ by $$\begin{aligned}
            f_i(\theta)&=\left|\operatorname{I\!I}\left(\cos (\theta)\,u+\sin (\theta)\,e_i,\cos (\theta)\,u+\sin (\theta)\,e_i\right)\right|^2\\
            &=\cos^4(\theta)|\operatorname{I\!I}(u,u)|^2+2\cos^2(\theta)\sin^2(\theta)\langle\operatorname{I\!I}(u,u),\operatorname{I\!I}(e_i,e_i)\rangle+\sin^4(\theta)|\operatorname{I\!I}(e_i,e_i)|^2.
        \end{aligned}$$ Then because $f_i(0)=|\operatorname{I\!I}(u,u)|^2$ is minimal, $$f_i''(0)=4\left(\langle\operatorname{I\!I}(u,u),\operatorname{I\!I}(e_i,e_i)\rangle-|\operatorname{I\!I}(u,u)|^2\right)\geq 0,$$ and so we must have $\langle\operatorname{I\!I}(u,u),\operatorname{I\!I}(e_i,e_i)\rangle\geq0$. Now let $\mathcal{V}=\mathrm{span}\{u,e_1,\dots,e_k\}\subseteq T_pN$. Because $\operatorname{I\!I}(u,e_i)=0$ and $\langle\operatorname{I\!I}(u,u),\operatorname{I\!I}(e_i,e_i)\rangle\geq0$ for all $i$, we have by the Gauss equation that $$\begin{aligned}
            \operatorname{Ric}_k(u,\mathcal{V})&=\sum_{i=1}^k\sec_M(u,e_i)\\
            &=\sum_{i=1}^k\left[\sec_N(u,e_i)+|\operatorname{I\!I}(u,e_i)|^2-\langle\operatorname{I\!I}(u,u),\operatorname{I\!I}(e_i,e_i)\rangle\right]\\
            &\leq \sum_{i=1}^k\sec_N(u,e_i)
        \end{aligned}$$ However, because $T_pN$ is spanned by commuting Killing fields, $\sec_N\equiv 0$ on $T_pN$. Therefore we have shown that if $\dim(N)>\frac{n+k}{2}$, then there exists a pair $(u,\mathcal{V})$ such that $\operatorname{Ric}_k(u,\mathcal{V})\leq 0$, thus proving Theorem \[thm:local\] by contraposition.

Construction of $k$-maximal local symmetry rank {#sec:example}
===============================================

In this section, we prove the following:

\[prop:models\] There exist metrics $g^\mathrm{model}$ on $\mathbb{R}^n$ that have $k$-maximal local symmetry rank at the origin for all $n\geq 3$ and $k\in\{1,\dots,n-2\}$.

Throughout this section, let $n\geq 3$, $k\in\{1,\dots,n-2\}$, and $d=\lfloor\frac{n+k}{2}\rfloor$. To prove Proposition \[prop:models\], we will construct metrics on $\mathbb{R}^{n-d}\times \mathbb{R}^d$ such that the coordinate vector fields for the $\mathbb{R}^d$ factor are Killing fields and $\operatorname{Ric}_k(u,\mathcal{V})>0$ for all $(k+1)$-dimensional subspaces $\mathcal{V}\subseteq T_0\mathbb{R}^d$. We do not claim that these metrics are complete as this will not be necessary for proving Theorem \[thm:dense\] in Section \[sec:close\]. First, we establish the following computational simplification:

\[lem:compsimp\] Given any Riemannian manifold $M$ and a natural number $d\leq\dim(M)-1$, suppose that $\{K_i\}_{i=1}^d$ is an orthonormal basis of a subspace $\mathcal{K}\subseteq T_pM$, $k\in\{1,\dots,d-1\}$, and there exist $\mu,\nu\in[0,\infty)$ such that the following hold:

1.  $R(K_i,K_j)K_l=0$ when the indices $i,j,l$ are mutually distinct. \[a\]

2.  $\sec(K_i,K_j)\in\{-\nu,\mu\}$ for all $i\neq j$. \[b\]

3.  For each $i$, there exist at most $k-1$ indices $j\neq i$ such that\
    $\sec(K_i,K_j)=-\nu$. \[c\]

4.  $\mu-(k-1)\nu>0$. \[d\]

Then $\operatorname{Ric}_k(u,\mathcal{V})>0$ for all $(k+1)$-dimensional subspaces $\mathcal{V}\subseteq\mathcal{K}$ and $u\in\mathcal{V}$.

Let $\operatorname{Ric}^\mathcal{K}:\mathcal{K}\to\mathcal{K}$ denote the Ricci $(1,1)$-tensor restricted to $\mathcal{K}$ and composed with projection onto $\mathcal{K}$. Then by definition, $\langle \operatorname{Ric}^\mathcal{K}(u),u\rangle=\operatorname{Ric}_{d-1}(u,\mathcal{K})$. By Property \[a\], we get that $\operatorname{Ric}^\mathcal{K}$ is diagonalized by $\{K_i\}_{i=1}^d$; see, for example, @petersen [Proposition 4.1.3]. Then by Properties \[b\] and \[c\], we have that $$\operatorname{Ric}_{d-1}(u,\mathcal{K})\geq \min_{i=1,\dots,d}\{\operatorname{Ric}_{d-1}(K_i,\mathcal{K})\}\geq(d-k)\mu-(k-1)\nu,\\$$ for all $u\in\mathcal{K}$. Applying Property \[d\], we get $$\operatorname{Ric}_{d-1}(u,\mathcal{K})>(d-k-1)\mu.
        \label{riccibound}$$

Now, define the operator $\mathfrak{R}^\mathcal{K}:\Lambda^2\mathcal{K}\to \Lambda^2\mathcal{K}$ by $\langle\mathfrak{R}^\mathcal{K}\left(\textstyle\sum_i X_i\wedge Y_i\right),\textstyle\sum_j V_j\wedge W_j\rangle=\sum_{i,j} R(X_i,Y_i,W_j,V_j)$, which is the curvature operator restricted to the subspace $\mathcal{K}$. Then by Property \[a\], $\mathfrak{R}^\mathcal{K}$ is diagonalized by $\{K_i\wedge K_j\}_{i,j}$; see, for example, @petersen [Proposition 4.1.2]. Recall that by Property \[b\], on $\mathcal{K}$ we have $$\sec\leq\mu.
        \label{secbound}$$ Thus, given a $(k+1)$-dimensional subspace $\mathcal{V}\subseteq\mathcal{K}$ and unit vector $u\in\mathcal{V}$, choose orthonormal vectors $\{e_i\}_{i=1}^{d-1}$ such that $\{u,e_1,\dots,e_k\}$ is an orthonormal basis for $\mathcal{V}$ and $\{u,e_1,\dots,e_k,e_{k+1},\dots,e_{d-1}\}$ is an orthonormal basis for $\mathcal{K}$. Then by Inequalities \[riccibound\] and \[secbound\], we have $$\begin{aligned}
    \operatorname{Ric}_k(u,\mathcal{V})&=\sum_{i=1}^k\sec(u,e_i)\\
    &=\operatorname{Ric}_{d-1}(u,\mathcal{K})-\sum_{i=k+1}^{d-1}\sec(u,e_i)\\
    &> \left[(d-k-1)\mu\right]-\left[(d-k-1)\mu\right]=0.
    \end{aligned}$$

Now we use Lemma \[lem:compsimp\] to construct metrics with $k$-maximal local symmetry rank:

\[Proof of Proposition \[prop:models\]\] Consider $\mathbb{R}^{n-d}\times \mathbb{R}^d$ with coordinates $(x_1,\dots,x_{n-d},y_1,\\\dots,y_d)$. Given positive smooth functions $\phi_i:\mathbb{R}^{n-d}\to\mathbb{R}$, define the metric $g^\mathrm{model}$ on $\mathbb{R}^{n-d}\times\mathbb{R}^d$ by $$g^\mathrm{model}={dx_1}^2+\dots+{dx_{n-d}}^2+{\phi_1}^2{dy_1}^2+\dots+{\phi_d}^2{dy_d}^2.$$ Setting $K_i=\frac{1}{\phi_{i}(0)}\frac{\partial}{\partial y_i}$, the fields $\{K_i\}_{i=1}^d$ are the desired commuting Killing fields under this metric, and they are orthonormal at the origin. We will choose the $\phi_i$ such that $\left(\mathbb{R}^n,g^\mathrm{model}\right)$ and $K_i$ together satisfy the hypotheses of Lemma \[lem:compsimp\].

Let $\mathcal{K}=\mathrm{span}\{K_i|_0\}_{i=1}^d=T_0\mathbb{R}^d$, and denote the orthogonal complement by $\mathcal{K}^\perp=T_0\mathbb{R}^{n-d}$. Let $\operatorname{I\!I}:\mathcal{K}\times\mathcal{K}\to \mathcal{K}^\perp$ denote the second fundamental form for the submanifold $\{(0,\dots,0)\}\times\mathbb{R}^d$. Then $\operatorname{I\!I}(K_i,K_j)=0$ for $i\neq j$, and combining this with the fact that $K_i$ are commuting Killing fields, we get that Property \[a\] of Lemma \[lem:compsimp\] is satisfied. We will show that the $\phi_i$ can be chosen such that Properties \[b\], \[c\], and \[d\] are satisfied at the origin. Notice that at the origin, $\operatorname{I\!I}(K_i,K_i)=-\frac{\operatorname{grad}\phi_i}{\phi_i(0)}$. Thus by the Gauss equation, because the submanifold $\{(0,\dots,0)\}\times\mathbb{R}^d$ has zero intrinsic curvature, we have that the extrinsic curvatures are $$\sec(K_i,K_j)=-\frac{\langle\operatorname{grad}\phi_i,\operatorname{grad}\phi_j\rangle}{\phi_i(0)\phi_j(0)}$$ when $i\neq j$. We now break the construction into two cases: $k=n-2$ and $k\leq n-3$.

**Case $\mathbf{k=n-2}$:** If $k=n-2$, then $d=\lfloor\frac{n+k}{2}\rfloor=n-1$, and so $\mathcal{K}^\perp$ is $1$-dimensional. Choose a unit vector $U$ in $\mathcal{K}^\perp$. We will define the $\phi_i$ such that for some constants $a,b\in(0,\infty)$,

1.  $\operatorname{grad}\phi_1=\dots=\operatorname{grad}\phi_k=U$,

2.  $\phi_1(0)=\dots=\phi_k(0)=a$,

3.  $\operatorname{grad}\phi_d=-U$, and

4.  $\phi_d(0)=b$

Then by construction, Properties \[b\] and \[c\] of Lemma \[lem:compsimp\] are satisfied with $$\begin{aligned}
    \mu&=\frac{|U|^2}{ab}=\frac{1}{ab},\\
    -\nu&=-\frac{|U|^2}{a^2}=-\frac{1}{a^2}.\end{aligned}$$ Thus, to satisfy Property \[d\], we need $$\frac{1}{ab}-\frac{k-1}{a^2}>0.$$ Therefore, choosing any values for $a$ and $b$ such that $a>(k-1)b$, the $k=n-2$ case of Proposition \[prop:models\] follows from Lemma \[lem:compsimp\].

**Case $\mathbf{k\leq n-3}$:** If $k \leq n-3$, then $\mathcal{K}^\perp$ has dimension $n-d\geq 2$. In the unit sphere $S^{n-d-1}\subset \mathcal{K}^\perp$, choose a vector $U$. Letting $U^\perp$ denote the orthogonal complement of $\mathrm{span}\{U\}$ in $\mathcal{K}^\perp$, consider the equatorial sphere $S^{n-d-2}=U^\perp\cap S^{n-d-1}$. Now inscribe a regular $(n-d-1)$-simplex in $S^{n-d-2}$, and define vectors $V_1(0),\dots,V_{n-d}(0)\in S^{n-d-2}$ to be the vertices of this simplex. Hence $$\langle V_i(0),V_j(0)\rangle=-\frac{1}{n-d-1}$$ for all distinct $i,j\in\{1,\dots,n-d\}$. Now for $\theta\in[0,\frac{\pi}{2}]$ define $$V_i(\theta)=\cos(\theta)V_i(0)-\sin(\theta)U.$$ Notice that $$\langle V_i(\theta),V_j(\theta)\rangle=-\frac{\cos^2\theta}{n-d-1}+\sin^2\theta$$ is strictly increasing on $[0,\frac{\pi}{2}]$ and takes the value $0$ at $\theta=\xi\in(0,\frac{\pi}{2})$ given by $$\xi=\arctan\left(\sqrt{\frac{1}{n-d-1}}\right).$$

Hence, $\langle V_i(\theta),V_j(\theta)\rangle$ is negative for $\theta\in[0,\xi)$. We will now choose the $\phi_i$ such that

1.  $\operatorname{grad}\phi_1=\dots=\operatorname{grad}\phi_k=U$, \[1\]

2.  $\phi_1(0)=\dots=\phi_k(0)=a$, \[2\]

3.  $\operatorname{grad}\phi_{k+j}=V_j(\theta)$ for $j=1,\dots,n-d$, and \[3\]

4.  $\phi_{k+1}(0)=\dots=\phi_d(0)=b$ \[4\]

for some values of $a,b\in(0,\infty)$ and $\theta\in(0,\xi)$. See Figure \[figure\] for an illustration of how the $\operatorname{grad}\phi_i$ may be arranged in $S^{n-d-1}$.

![$\operatorname{grad}\phi_{i}$ in $S^{n-d-1}$<span data-label="figure"></span>](gradients.eps){width="60.00000%"}

Then because $\theta<\xi$, the negative curvatures among $\{\sec(K_i,K_j)\}_{i,j=1}^d$ correspond to distinct values of $i,j\in\{1,\dots,k\}$, and these curvatures all have the same value $$-\nu{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
            \hbox{\scriptsize.}\hbox{\scriptsize.}}}    =}-\frac{|U|^2}{a^2}=-\frac{1}{a^2}.$$ So for each $i$, there exist at most $k-1$ indices $j\neq i$ such that $\sec(K_i,K_j)=-\nu$, and thus Property \[c\] of Lemma \[lem:compsimp\] is satisfied. To satisfy Property \[b\], we need to choose $a,b,\theta$ so that for all $l\in\{1,\dots,k\}$ and for all distinct $i,j\in\{1,\dots,n-d\}$, $$\sec(K_l,K_{k+i})=\sec(K_{k+i},K_{k+j}).$$ This common value is given by $$\mu{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
            \hbox{\scriptsize.}\hbox{\scriptsize.}}}    =}-\frac{\langle U,V_i(\theta)\rangle}{ab}=-\frac{\langle V_i(\theta),V_j(\theta)\rangle}{b^2}.
    \label{mu}$$ Furthermore, to satisfy Property \[d\], we need $$\mu-\frac{k-1}{a^2}>0.
    \label{ineqd}$$ Now choose any value for $a>0$. Then, once $\theta\in(0,\xi)$ is chosen, we will define $b$ by $$b=\frac{a\langle V_i(\theta),V_j(\theta)\rangle}{\langle U,V_i(\theta)\rangle}.$$ Thus Equation \[mu\] will be satisfied with $\mu$ taking the value $$\mu=-\frac{\langle U,V_i(\theta)\rangle^2}{a^2\langle V_i(\theta),V_j(\theta)\rangle}.$$ Finally, Inequality \[ineqd\] holds if $\theta$ is chosen such that $$\langle U,V_i(\theta)\rangle^2>-(k-1)\langle V_i(\theta),V_j(\theta)\rangle.$$ Notice that $\langle U,V_i(\theta)\rangle^2$ approaches a positive constant dependent on $n$ and $k$ as $\theta$ approaches $\xi$, while $-\langle V_i(\theta),V_j(\theta)\rangle$ approaches $0$ as $\theta$ approaches $\xi$. Therefore, there exists a value $\theta$ such that Inequality \[ineqd\] holds. Therefore, applying Lemma \[lem:compsimp\], Proposition \[prop:models\] is proven.

Sewing and the $C^1$-topology {#sec:close}
=============================

In this section, we establish a general Sewing Theorem for changing a Riemannian metric within metric ball around a point while remaining close to the original metric in the $C^1$-topology. We then use the Sewing Theorem to Prove Theorem \[thm:dense\]. First, we discuss the $C^l$-norm for tensors.

$C^l$-norm for tensors
----------------------

Recall that two smooth maps $F_1,F_2:M\to N$ are $\epsilon$-close in the <span>*weak $C^l$-topology*</span> if their values and partial derivatives up to order $l$ are $\epsilon$-close with respect to fixed atlases on $M$ and $N$; see Chapter 2 of @hirsch. If the atlases are finite, this leads to a notion of <span>*$C^l$-distance*</span>.

Now, given vector bundles $E_1$ and $E_2$, Euclidean metrics on $E_1$ and $E_2$, and a bundle map $\phi:E_1\to E_2$, the <span>*$C^l$-norm*</span> of $\phi$ is defined as follows: If $E_1^1$ denotes the unit sphere bundle under the Euclidean metric on $E_1$, then $|\phi|_{C^l}$ is the $C^l$-distance from $\phi|_{E_1^1}$ to the zero bundle map $E_1\to E_2$. The $C^l$-norm of a tensor $\omega$ is the $C^l$-distance from $\omega$ to the zero-section. Notice that these definitions depend on the choice of Euclidean metrics.

Throughout this section, let $M$ be a fixed manifold, and let $g$ be a fixed Riemannian metric on $M$. All $C^l$-norms will be defined in terms of the fixed metric $g$ on $TM$.

Sewing theorem
--------------

We now establish the following:

\[Sewing Theorem\]\[thm:sewing\] Let $(M,g)$ be a Riemannian manifold, let $p$ be a point in $M$, and let $g^*$ be a Riemannian metric defined on a neighborhood of $p$ such that $g(\gamma',\cdot)=g^*(\gamma',\cdot)$ for all geodesic rays $\gamma$ emanating from $p$. For every $\epsilon>0$, there exists a Riemannian metric $\tilde{g}$ on $M$ that is $\epsilon$-close to $g$ in the $C^1$-distance such that $\tilde{g}\equiv g^*$ on an open ball centered at $p$.

The author has been informed that Searle, Solórzano, and Wilhelm have proven if $\sec(M,g)\geq K$, then $\tilde{g}$ in Theorem \[thm:sewing\] can be made to satisfy $\sec(M,\tilde{g})\geq \tilde{K}$ for any $\tilde{K}< K$ @SSWprep.

Now define $t$ to be the value of $\operatorname{dist}_{g}(p,\cdot)$, and let $\partial_t$ denote $\operatorname{grad}(\operatorname{dist}_g(p,\cdot))$. We begin by proving the following:

\[Converse Gauss Lemma\]\[convgauss\] Let $(M,g)$ be a Riemannian manifold, let $p\in M$, and let $g^*$ be as in Theorem \[thm:sewing\].

1.  $\operatorname{dist}_g(p,\cdot)=\operatorname{dist}_{g^*}(p,\cdot)$, and the integral curves of $\partial_t$ are geodesics emanating from $p$ with respect to both metrics $g$ and $g^*$. \[prop1\]

2.  Along geodesic rays emanating from $p$, the family of Jacobi fields that vanish at $p$ are the same for both metrics $g$ and $g^*$. \[prop2\]

For a more general version of Lemma \[convgauss\], see @princdeform [Proposition 2.2].

Because $g^*(\partial_t,\partial_t)=g(\partial_t,\partial_t)=1$, the integral curves of $\partial_t$ are also geodesics under $g^*$, and hence Part (\[prop1\]) follows. Now notice that along a geodesic ray emanating from $p$, a Jacobi field $J$ that vanishes at $p$ under $g$ is realized as the variation field of a variation by $g$-geodesics emanating from $p$. Because these curves are also geodesics under $g^*$, $J$ is also a Jacobi field under $g^*$.

Now letting $\operatorname{inj}\operatorname{rad}_p$ denote the injectivity radius of $(M,g)$ at $p$, choose $\delta\in\\(0,\frac{1}{2}\operatorname{inj}\operatorname{rad}_p)$ small enough such that $g^*$ is defined on the closed ball $\overline{B(p,2\delta)}\subset M$.

\[lem:r\] There is a symmetric $(0,2)$-tensor $r$ defined on $\overline{B(p,2\delta)}$ such that $$g-g^*=t^2r.$$ Furthermore, for any $l\in\mathbb{N}$, there exists a constant $C$ such that $|r|_{C^l}<C$ with respect to the metric $g$.

Choose normal coordinates $(x^1,\dots,x^n)$ around $p$, and let $\partial_1,\dots,\partial_n$ denote the corresponding coordinate vector fields. Consider the rotational fields $J=x^j\partial_i-x^i\partial_j$. These fields are normal to $\partial_t=\operatorname{grad}(\operatorname{dist}(p,\cdot))$, and they vanish at $p$. Now consider any geodesic $\gamma:[0,2\delta]\to M$ emanating from $p$. Then $J$ are also Jacobi fields along $\gamma$ because the Lie derivative $\mathcal{L}_{\partial_t}J=[\partial_t,J]=0$. Notice that for every point $\gamma(t_0)$ with $t_0>0$, there is a neighborhood around $\gamma(t_0)$ such that every tangent space is spanned by $\partial_t$ and $n-1$ of the rotational fields $J$. We choose $n-1$ such rotational fields and denote them by $J_1,\dots,J_{n-1}$. We will construct the tensor $r$ in this neighborhood of $\gamma(t_0)$ by defining its values on $\big\{\frac{J_1}{|J_1|},\dots,\frac{J_{n-1}}{|J_{n-1}|},\partial_t\big\}$. By the Gauss Lemma, $(g-g^*)(\partial_t,\cdot)\equiv0$, and so we set $r(\partial_t,\cdot)\equiv0$. Informed by @docarmo [Chapter 5 Proposition 2.7], we now compute the Taylor expansion of $(g-g^*)(J_i,J_j)_{\gamma(t)}$ centered at $t=0$, where $\gamma$ is a geodesic emanating from $p$. First notice that because the $J_i$ are Jacobi and $J_i(0)=0$, we have $$J_i''(0)=-\left(R_g(J_i,\partial_t)\partial_t\right)(0)=0.$$ Second, for any vector field $V$ along $\gamma$, we have at $t=0$ $$\begin{aligned}
            g\left(\nabla_{\partial_t}(R_g(J_i,\partial_t)\partial_t),V\right)&=\partial_tg\left(R_g(J_i,\partial_t)\partial_t,V\right)-g\left(R_g(J_i,\partial_t)\partial_t,V'\right)\\
            &=\partial_tg\left(R_g(V,\partial_t)\partial_t,J_i\right)\\
            &=g\left(\nabla_{\partial_t}(R_g(V,\partial_t)\partial_t),J_i\right)+g\left(R_g(V,\partial_t)\partial_t,J_i'\right)\\
            &=g\left(R_g(J_i',\partial_t)\partial_t,V\right).
        \end{aligned}$$ Using these facts and that $J_i(0)=0=J_j(0)$, we have the following at $t=0$: $$\begin{aligned}
            g(J_i,J_j)&=0,\\[2ex]
            g(J_i,J_j)'&=g(J_i',J_j)+g(J_i,J_j')=0,\\[2ex]
            g(J_i,J_j)''&=g(J_i'',J_j)+2g(J_i',J_j')+g(J_i,J_j'')=2g(J_i',J_j'),\\[2ex]
            g(J_i,J_j)'''&=g(J_i''',J_j)+3g(J_i'',J_j')+3g(J_i',J_j'')+g(J_i,J_j''')=0,\\[2ex]
            g(J_i,J_j)''''&=g(J_i'''',J_j)+4g(J_i''',J_j')+6g(J_i'',J_j'')+4g(J_i',J_j''')+g(J_i,J_j'''')\\
            &=-8R_g(J_i',\partial_t,\partial_t,J_j').
        \end{aligned}$$ By Lemma \[convgauss\], $\gamma$ is a geodesic for both $g$ and $g^*$, and $J_i,J_j$ are Jacobi fields along $\gamma$ for either metric. Hence, we have that the equations above also hold with $g^*$ substituted for $g$. Thus, by applying these calculations to the Taylor expansions of $g(J_i,J_j)_{\gamma(t)}$ and $g^*(J_i,J_j)_{\gamma(t)}$ centered at $t=0$, and using the fact that the inner products induced by $g$ and $g^*$ on $T_pM$ agree, we have $$(g-g^*)(J_i,J_j)_{\gamma(t)}=-\frac{t^4}{3}(R_g-R_{g^*})(J_i',\partial_t,\partial_t,J_j')_p+O(t^5)\qquad \text{as } t\to 0$$ By normalizing this equation, we get $$(g-g^*)\left(\frac{J_i}{|J_i|},\frac{J_j}{|J_j|}\right)_{\gamma(t)}=\frac{t^4}{3|J_i(t)||J_j(t)|}(R_{g^*}-R_g)(J_i',\partial_t,\partial_t,J_j')_p+O(t^3),$$ where $|J_i|$ is measured using $g$. It follows from the computations of derivatives $g(J_i,J_j)$ above that $|J_i(t)|=t+O(t^2)=|J_j(t)|$ as $t\to 0$. Thus, we have $$\label{eq:derivative}
            \lim_{t\to 0}\frac{d}{dt}\left[(g-g^*)\left(\frac{J_i}{|J_i|},\frac{J_j}{|J_j|}\right)_{\gamma(t)}\right] = 0.$$ Now, recall that if a smooth function $f:\mathbb{R}\to\mathbb{R}$ satisfies $f(0)=0=f'(0)$, then there is a smooth function $h:\mathbb{R}\to\mathbb{R}$ such that $f(t)=t^2h(t)$. Thus $$(g-g^*)\left(\frac{J_i}{|J_i|},\frac{J_j}{|J_j|}\right)_{\gamma(t)}=t^2r_{i,j}(t)$$ for some smooth function $r_{i,j}$. Thus we define $r\big(\frac{J_i}{|J_i|},\frac{J_j}{|J_j|}\big){\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
            \hbox{\scriptsize.}\hbox{\scriptsize.}}}    =}r_{i,j}$ so that $g-g^*=t^2r$. Because the inner products induced by $g$ and $g^*$ on $T_pM$ agree, it follows from Equation \[eq:derivative\] and Taylor's theorem that $r$ can be smoothly extended to $\overline{B(p,2\delta)}$. Finally, the inequality $|r|_{C^l}<C$ follows from the fact that $r$ is smooth on the compact set $\overline{B(p,2\delta)}$.

We are now prepared to prove the Sewing Theorem.

\[Proof of Theorem \[thm:sewing\]\] Choose $\delta\in(0,\frac{1}{2}\operatorname{inj}\operatorname{rad}_p)$ such that $g^*$ is defined on the closed ball $\overline{B(p,2\delta)}\subset M$. By Lemma 7.3 in @diffstab, there exists a smooth function $\phi:M\to [0,1]$ such that $$\begin{aligned}
            &\phi\equiv 1 \text{ where }\operatorname{dist}(p,\cdot)\leq \delta,\\
            &\phi\equiv 0 \text{ where } \operatorname{dist}(p,\cdot)\geq 2\delta,\\
            |&\phi|_{C^1}\leq \frac{2}{\delta}.
        \end{aligned}$$

On $M$, define the Riemannian metric $\tilde{g}=(1-\phi)g+\phi g^*$. Because $\tilde{g}\equiv g$ for $\operatorname{dist}(p,\cdot)\geq 2\delta$, assume $t=\operatorname{dist}(p,\cdot)<2\delta$. Given unit vector fields $U$ and $V$, $$\begin{aligned}
            (g-\tilde{g})(U,V)&=g(U,V)-(1-\phi)g(U,V)-\phi g^*(U,V)\\
            &=\phi (g-g^*)(U,V)\\
            &=\phi t^2 r(U,V),
        \end{aligned}$$ where $r$ is the tensor from Lemma \[lem:r\]. Furthermore, for a unit vector $X$, we have $$\begin{aligned}
            X(g-\tilde{g})(U,V)&=(X\phi) t^2r(U,V)+\phi X(t^2)r(U,V)+\phi t^2Xr(U,V)\\
            &=(X\phi) t^2r(U,V)+\phi(2t)(Xt)r(U,V)+\phi t^2Xr(U,V).
        \end{aligned}$$ Thus, because $\phi\leq 1$, $|\phi|_{C^1}\leq \frac{2}{\delta}$, $t<2\delta$, and $|r|_{C^1}<C$, we have $$\begin{aligned}
            |(g-\tilde{g})(U,V)|&<4\delta^2C,\\
            |X(g-\tilde{g})(U,V)|&<12\delta C+4\delta^2C.
        \end{aligned}$$ Therefore, for any $\epsilon>0$, $\delta$ can be chosen so that $|g-\tilde{g}|_{C^1}<\epsilon$.

Density of $k$-maximal local symmetry rank
------------------------------------------

In this section, we prove Theorem \[thm:dense\] using Theorem \[thm:sewing\] and the metrics established in Proposition \[prop:models\].

\[Proof of Theorem \[thm:dense\]\] Choose any Riemannian $n$-manifold $(M,g)$ with $n\geq 3$. Let $p\in M$ and define $g^*$ on a neighborhood of $p$ to be the pull-back metric $$g^*=\left(\exp_p^{-1}\right)^*\left(g^\mathrm{model}\right),$$ where $g^\mathrm{model}$ is one of the metrics from Proposition \[prop:models\] in Section \[sec:example\]. Then by the Gauss Lemma, $g(\gamma',\cdot)=g^*(\gamma',\cdot)$ for all geodesic rays $\gamma$ emanating from $p$, and thus Theorem \[thm:sewing\] can be applied to obtain a metric $\tilde{g}$ on $M$ that has $k$-maximal local symmetry rank at $p$. Therefore, the space of metrics on $M$ which have $k$-maximal local symmetry rank is dense under the $C^1$-topology.

{'source': 'arXiv_out/1901'}