On Riemannian submersions

TL;DR

This paper proves that the image of a real analytic Riemannian manifold under a smooth Riemannian submersion is necessarily real analytic, establishing a key property of such mappings.

Contribution

It establishes that the image of a real analytic Riemannian manifold under a smooth Riemannian submersion is always real analytic, a new result in differential geometry.

Findings

The image of a real analytic Riemannian manifold under a smooth Riemannian submersion is real analytic.

The result bridges properties of the manifold and its image under submersion.

Provides foundational insight into the structure of Riemannian submersions.

Abstract

We prove that the image of a real analytic Riemannian manifold under a smooth Riemannian submersion is necessarily real analytic.

Equations

$$\{a+ib:a,b\in \mathbb{R},|a|+|b|r\le r\},$$

$$M\times G_r\ni (x,g)\mapsto xg\in M$$

$$G_r\ni g\mapsto xg\in M$$

$$M_r\ni x\mapsto \dot{x}(t)\in TX$$

$$M_r\times G_r\ni (x,g)\mapsto xg=x\circ g\in M_r,$$

$$M_r\ni x\mapsto \dot{x}(0)\in TX,N_r\ni y\mapsto \dot{y}(0)\in TY,$$

$$H=(\text{Ker}{\phi }_{\ast }{)}^{\perp }\subset TX,$$

$$0=(d{\psi }^{\ast }\alpha )|P_x=({\psi }^{\ast }d\alpha )|P_x,$$

$${\Omega }_{\psi (x)\ast }S={\psi }_{\ast }{\Omega }_{x\ast }S\subset {\psi }_{\ast }P=Q,$$

$$\alpha (\xi )=h(v,{\pi }_{\ast }\xi ),\xi \in T_{v}TX,\text{and}\omega =d\alpha .$$

$$h(v,u)=h(v,u_H)=k({\phi }_{\ast }v,{\phi }_{\ast }{u}_H)=k({\phi }_{\ast }v,{\phi }_{\ast }u).$$

$$\beta ({\Phi }_{\ast }\xi )=k(\Phi v,{\varrho }_{\ast }{\Phi }_{\ast }\xi )=k({\phi }_{\ast }v,{\phi }_{\ast }{\pi }_{\ast }\xi )=h(v,{\pi }_{\ast }\xi )=\alpha (\xi ).$$

$${\psi }^{\ast }\sigma =\omega |H_r.$$

$$\text{Ker}{\psi }_{\ast }^{\mathbb{R}}=\{\xi \in T_x{H}_r:x\in H_r\text{ and }\omega (\xi ,\eta )=0\text{ for all }\eta \in T_x{H}_r\}.$$

$$J\text{Ker}{\psi }_{\ast }^{\mathbb{R}}\subset T(M\cap H_r{)}^{\perp }.$$

$$\dim M_r=2\dim X,\dim N_r=2\dim Y,\dim H_r=\dim X+\dim Y,$$

$$\text{rk}T(M\cap H_r{)}^{\perp }=\dim X-\dim Y,\text{rk}\text{Ker}{\psi }_{\ast }^{\mathbb{R}}=\dim H_r-\dim N_r=\dim X-\dim Y,$$

$$E=(\text{Ker}{\psi }_{\ast }^{\mathbb{R}}\oplus J\text{Ker}{\psi }_{\ast }^{\mathbb{R}}{)}^{\perp }$$

$$\{\xi -iJ\xi :\xi \in E\}$$

$${\psi }_{\ast }{R}_x\oplus {\psi }_{\ast }{\overline{R}}_x=\mathbb{C}{T}_{\psi (x)}N.$$

$$E=\{\eta \in T_x{H}_r:x\in M\cap H_r\text{ and }\omega (\xi ,\eta )=0\text{ for all }\xi \in (\text{Ker}{\psi }_{\ast }{)}_x\}$$

$$G_r\ni g\mapsto ({\xi }_{\alpha }g{)}^{\prime },({\eta }_{\alpha }g{)}^{\prime }\in P=T^{10}M$$

$$({\eta }_{\alpha }g{)}^{\prime }=\sum _{\beta =1}^m{F}_{\alpha \beta }(g)({\xi }_{\beta }g{)}^{\prime },\alpha =1,\dots ,n,g\in \text{int}{G}_r.$$

$${\psi }_{\ast }({\xi }_{\alpha }g),{\psi }_{\ast }({\eta }_{\alpha }g)\in T_{xg}Y,g\in \Gamma .$$

$${\eta }_{\alpha }g=\sum _{\beta =1}^n{f}_{\alpha \beta }(g){\xi }_{\beta }g,\text{ or equivalently, }{\psi }_{\ast }{\eta }_{\alpha }g=\sum _{\beta =1}^n{f}_{\alpha \beta }(g){\psi }_{\ast }{\xi }_{\beta }g,$$

$${\eta }_{\alpha }^{\prime }=\sum _{\beta =1}^n{F}_{\alpha \beta }(e){\xi }_{\beta }^{\prime },\alpha =1,\dots ,n.$$

Full text

1 Introduction

A $C^{1}$ map $\varphi$ between Riemannian manifolds $(X,h)$ and $(Y,k)$ is a Riemannian submersion if for every $x\in X$ the differential $\varphi_{*}$ induces the metric of $T_{\varphi(x)}Y$ from the metric of $T_{x}X$. In other words, $\varphi_{*}$ maps the orthogonal complement of Ker$\,\varphi_{*}$ isometrically on $TY$. Our main theorem is the following regularity result.

Theorem 1.1**.**

Suppose $(X,h)$ is a real analytic and $(Y,k)$ is a $C^{\infty}$ Riemannian manifold, both finite dimensional. If $\varphi:(X,h)\to(Y,k)$ is a surjective $C^{\infty}$ Riemannian submersion, then $Y$ admits a real analytic structure in which $k$ is real analytic.

Henceforward we will drop “real” from real analytic.—An amusing aspect of the theorem is that $\varphi$ itself need not be analytic. Let $(X,h)$ be an analytic Riemannian manifold, and $S\subset X$ a smooth hypersurface that cuts $X$ in two. Then $\varphi(x)=\pm\text{dist}\,(x,S)$, $x\in X$, in a neighborhood of $S$ satisfies $|d\varphi|\equiv 1$, and so defines a Riemannian submersion to $\mathbb{R}$, which fails to be analytic unless $S$ is.

In Theorem 4.1 we will prove a more precise result, with $(Y,k)$ only $C^{p}$. The proof will use so called adapted complex structures. These structures first appeared in [LSz1] as complex manifold structures on the tangent bundle of Riemannian manifolds $(X,h)$ that are in a sense compatible with the metric. Simultaneously, Guillemin and Stenzel in [GS] discovered an equivalent complex structure, but on the cotangent bundle. Guillemin–Stenzel and Szőke in [Sz1, Sz3] prove that if $(X,h)$ is analytic, then on some neighborhood of the zero section in $T^{*}X$, resp. $TX$, an adapted complex structure exists; and this author in [L1] proved the converse. Accordingly, we will derive Theorem 1.1 from a result that connects the existence of adapted complex structures for a Riemannian manifold and for its image under a Riemannian submersion.

It turns out that it is better to approach this latter result from a perspective somewhat different from [GS, LSz1], and view adapted complex structures not on $TX$ or on $T^{*}X$, but on the related manifold $M_{r}$ of geodesics $x:[-r,r]\to X$, as suggested in [LSz2]. Here $r$ is a positive number. For one thing, a Riemannian submersion $\varphi:X\to Y$ induces a map $\varphi_{*}:TX\to TY$, but $\varphi_{*}$ cannot immediately be employed to transfer adapted complex structures from $TX$ to $TY$. Working with manifolds $M_{r},N_{r}$ of geodesics in $X,Y$ though, it becomes obvious that $\varphi$ does induce a submersion $\psi$ from a submanifold $H_{r}\subset M_{r}$ to $N_{r}$; namely from the submanifold of geodesics in $X$ perpendicular to the fibers of $\varphi$. Now $H_{r}$ is not a complex submanifold of $M_{r}$. However, an adapted complex structure of $M_{r}$, wherever it exists, restricts to $H_{r}$ as a CR structure. This suggests that we enlarge the notion of adapted complex structures, and develop a quotient construction in this more general framework, that ultimately yields an adapted complex structure on $N^{r}$. We have already written in [L2] on the generalization in question, a variant of which we will review in the next section. In Section 3 we discuss how an adapted complex structure associated with $(X,h)$ induces an adapted complex structure associated with its image $(Y,k)$ under a Riemannian submersion, and in Section 4 we prove Theorem 1.1 in a more precise form.

2 Adapted involutive structures

Here we will review the notion of involutive structures adapted to the action of a certain monoid, a special instance of the notion studied in [L2]. In one aspect we will be more general, though: while [L2] worked with $C^{\infty}$ and more regular objects, in this paper we will work with $C^{p}$ Hölder classes, $p\in[0,\infty]$. At the same time, unlike in [L2], all our manifolds will be finite dimensional. If $E$ is a vector bundle, we abbreviate $\mathbb{C}\otimes_{\mathbb{R}}E$ as $\mathbb{C}E$.

Definition 2.1**.**

Let $p\geq 1$ and $M$ a manifold of class $C^{p+1}$. An involutive structure of class $C^{p}$ on $M$ is a subbundle $P\subset\mathbb{C}TM$ of class $C^{p}$ such that the Lie bracket of its $C^{1}$ sections is again a section. (Such a bundle itself is also called involutive.) A $C^{1}$ map $f:(M,P)\to(N,Q)$ of involutive manifolds is involutive if $f_{*}P\subset Q$.

For example, if $P\oplus\overline{P}=\mathbb{C}TM$, $P$ is an integrable almost complex structure. According to Hill and Taylor [HT], when $p>1$ is not an integer, one can find $\mathbb{C}$–valued local coordinates $z_{1},\dots,z_{m}$ on $M$, of class $C^{p+1}$, such that $dz_{j}|\overline{P}=0$. The case $p=\infty$ was first proved in [NN]. The local coordinates turn $M$ into a complex manifold, whose bundle of $(1,0)$ vectors is $T^{10}M=P$. More generally, when $P\cap\overline{P}=(0)$, $P$ defines a CR structure, that may or may not be embeddable in a complex manifold.

While Definition 2.1 is made for manifolds $M$ without boundary, it makes sense even for manifolds with corners. We will need involutive structures on one type of manifolds with corners: Fix $r\in(0,\infty)$ and consider affine maps $\mathbb{R}\ni t\mapsto a+bt\in\mathbb{R}$ that map $[-r,r]$ into itself. These maps form a monoid $G_{r}$, with the operation composition of maps. Let $e\in G_{r}$ denote the neutral element. The coordinates $a,b$ identify $G_{r}$ with

[TABLE]

a diamond in $\mathbb{C}$. We endow $G_{r}$ with the complex structure $S=T^{10}\mathbb{C}|G_{r}$.

Definition 2.2**.**

Let $M$ be a manifold of class $C^{p}$. A right action of $G_{r}$ on $M$, of class $C^{p}$, is a $C^{p}$ map

[TABLE]

such that $(xg)h=x(gh)$ and $xe=x$.

Definition 2.3**.**

Given a $C^{1}$ right action of $G_{r}$ on $M$, an involutive structure $P$ on $M$ is adapted (to $G_{r}$) if for every $x\in M$

[TABLE]

is an involutive map between $({\rm{int}}\,G_{r},S)$ and $(M,P)$.

Consider next a manifold $X$ of class $C^{p+1}$, $p\geq 2$, endowed with a Riemannian metric $h$ of class $C^{p}$. Given $r\in(0,\infty)$, let $M_{r}$ be the set of geodesics $x:[-r,r]\to X$ of $h$. For any $t\in[-r,r]$

[TABLE]

is a bijection on an open subset of $TX$, and endows $M_{r}$ with a $C^{p}$ manifold structure, that is independent of the choice of $t$. If $(X,h)$ was analytic, so will be $M_{r}$. In [LSz1, LSz2] we focused on complete $(X,h)$ and the manifold $M_{\infty}$ of all geodesics $x:\mathbb{R}\to X$. Of course, then all $M_{r}$ are the same, in the sense that the map $M_{\infty}\ni x\mapsto x|[-r,r]\in M_{r}$ is a diffeomorphism. In general, we embed $X$ into $M_{r}$ by associating with $u\in X$ the constant geodesic $x\equiv u$, and view $X\subset M_{r}$ as a $C^{p}$ submanifold.

The monoid $G_{r}$ acts on $M_{r}$ on the right:

[TABLE]

and so on $G_{r}$ invariant submanifolds of $M_{r}$ we can talk about complex or involutive structures that are adapted to $G_{r}$. The connection with adapted complex structures in the sense of [LSz1, Sz3] is as follows. Embed $M_{r}$ into $TX$ by $x\mapsto\dot{x}(0)$. The action of $g(t)=a+bt$ on $M_{r}$ corresponds to following the geodesic flow on $TX$ for time $a$ and then fiberwise dilating by a factor of $b$. Note that the geodesic flow, as the flow of a $C^{p-1}$ vector field, is $C^{p-1}$. The dilation action is also, hence the corresponding action (2.1) is $C^{p-1}$. When $(X,h)$ is complete and $U$ is a neighborhood of the zero section in $TX=M_{\infty}$, [LSz1, Definition 4.1] defined a complex structure on $U$ adapted if the image $ug$ of any $u\in U$ under this map depends holomorphically on $a+bi$, and [Sz3] extended this to geodesics of a not necessarily complete connection. This implies that under the embedding $M_{r}\hookrightarrow TX$ adapted complex structures in the sense of [LSz1, Sz3] and of Definition 2.3 correspond.

The following is a special case of a result of Szőke, [Sz3, Theorem 0.3]. It can also be derived from [GS].

Theorem 2.4**.**

If $(X,h)$ is an analytic Riemannian manifold and $r\in(0,\infty)$, any constant geodesic $x\equiv u$ in $X$ admits a $G_{r}$ invariant neighborhood in $M_{r}$ and an analytic complex structure $P$ on it that is adapted to $G_{r}$.

Analyticity is not only sufficient but necessary for the existence of adapted complex structures:

Theorem 2.5**.**

Suppose $p>3$ is not an integer and $Y$ is a manifold of class $C^{p+1}$ endowed with a Riemannian metric $k$ of class $C^{p}$. Given $r\in(0,\infty)$, let $N_{r}$ stand for the $C^{p}$ manifold of geodesics $y:[-r,r]\to Y$, into which $Y$ is embedded as the submanifold of constant geodesics. If a $G_{r}$ invariant neighborhood of $Y\subset N_{r}$ admits an adapted complex structure $P$ of class $C^{p-1}$, then there is an analytic manifold structure on $Y$ in which $k$ is analytic, and whose underlying $C^{p}$ manifold is the same as that of the initial $C^{p+1}$ manifold. The same holds if $p\geq 3$ is an integer, with the sole difference that instead of the underlying $C^{p}$ manifolds only the agreement of the underlying $C^{p-\varepsilon}$ manifold structures can be guaranteed, with arbitrary $\varepsilon>0$.

Proof.

When $(Y,k)$ is compact and $p=\infty$, we proved this in [L1], see Theorem 1.5 there. We indicate the modifications needed for the more general and precise statement. Let $U\subset N_{r}$ be a neighborhood of $Y$ on which the adapted complex structure $P$ exists. According to Hill and Taylor [HT] locally on $U$ there are $C^{p}$ (or $C^{p-\varepsilon}$ if $p$ is an integer) functions $z_{1},\dots,z_{n}$ that serve as holomorphic coordinates $(d\overline{z}_{j}|P=0)$. This endows $U$ with the structure of a complex manifold whose underlying $C^{p}$ (resp. $C^{p-\varepsilon}$) manifold agrees with that of the original structure of $U\subset TY$. Let $y^{*}(t)=y(-t)$. As in [LSz1, Theorem 5.7] it follows that $y\mapsto y^{*}$ is an antiholomorphic map in a neighborhood of $Y$. Hence $Y$, its fixed point set, is an analytic submanifold of the complex manifold $U$. What remains is to prove that $k$ is analytic in the inherited analytic structure of $Y$. [L1] does this by passing from the real quadratic form $k$ on $TY$ to the induced complex quadratic form $q$ on $T^{10}U|Y$ and producing a holomorphic extension of $q$ to $T^{10}U_{1}$, with $U_{1}\subset U$ a neighborhood of $Y$.

The extension is constructed via certain $n$ by $n$ matrices $f=(f_{\alpha\beta})$ of meromorphic functions associated with geodesics $y\in N_{r}$. In [L1] the geodesics extend to complete geodesics $\mathbb{R}\to Y$ and the $f_{\alpha\beta}$ are meromorphic on strips in $\mathbb{C}$ about $\mathbb{R}$, of width inversely proportional to the speed of the geodesic. In the case at hand all we can say is that if we stay in a sufficiently small neighborhood $V\subset U$ of a fixed constant geodesic $y_{0}\in Y\subset N_{r}$, any $y\in V$ will extend to a geodesic on an arbitrarily large interval $[-R,R]$. By choosing $V$ suitably we can arrange that the $f_{\alpha\beta}$ associated with $y\in V$ are meromorphic on the disc $D=\{s\in\mathbb{C}:|s|<1\}$, with poles only on $\mathbb{R}\cap D$. [LSz1, Lemma 6.7] generalizes to the current setting, and gives that $\pm\text{Im}\,f$ is positive definite on $D^{\pm}=\{s\in D:\pm\text{Im}\,s>0\}$. (An alternative proof is in [Sz2], cf. (4.5) there.) Further, $f(0)=0$.

To construct the holomorphic extension of $q$ to $T^{10}V_{1}$ with some neighborhood $V_{1}\subset V$ of $y_{0}$ we need a neighborhood of $0\in D$ free of zeros of $\det f^{\prime}$, cf. Lemma 3.2 in [L1]. Fix $\sigma\in(0,1)$ so that conjugate points along any $y\in V$ are at distance $>2\sigma$. As in [L1], this implies that the pole of $f^{-1}$ closest to the pole at 0, is at distance $>2\sigma$. Instead of finding a zero free region about 0 we will find one about $\sigma$. In view of the transformation formula (2.5) and Lemma 2.4 of [L1], this neighborhood can be transformed back to 0.

Now $f^{\prime}(s)$ is invertible if $s\in(0,2\sigma)$ is not a pole of $f$ by the same argument as in [L1, Lemma 3.2]. For $s\in D^{+}$ it is easier to study the invertibility of $(f^{-1}){{}^{\prime}}(s)$. One pushes $f^{-1}$ forward to a holomorphic matrix function $F$ on the upper half plane $\mathbb{C}^{+}$ by a fixed conformal map $\varphi:D^{+}\to\mathbb{C}^{+}$. [L1, Proposition 3.1] guarantees a neighborhood of $\varphi(\sigma)$ in $\overline{\mathbb{C}^{+}}$ where $F^{\prime}$ is invertible, hence a neighborhood of $\sigma$ in $\overline{D^{+}}$ where $(f^{-1}){{}^{\prime}}$ is invertible. The neighborhood is independent of $y\in V$. Replacing $D^{+}$ by $D^{-}$ we in fact obtain a neighborhood of $\sigma\in D$, independent of $y\in V$, where $0\neq\det(f^{-1})^{\prime}=\det f^{\prime}/(\det f)^{2}$. The upshot is that on a fixed neighborhood of $0\in D$ the matrices $f$ associated with geodesics $y\in V$ have det$\,f^{\prime}\neq 0$ away from the poles of $f$. Once this granted, the rest of the proof in [L1] goes through, and provides the holomorphic extension of $q$, whence the analyticity of $h$.

3 Riemannian submersions

Now we come to the main point of the paper. Let $p\in[2,\infty]$, $X$ and $Y$ $C^{p+1}$ manifolds endowed with $C^{p}$ metrics $h,k$ and $\varphi:(X,h)\to(Y,k)$ a surjective Riemannian submersion of class $C^{p+1}$. Given $r\in(0,\infty)$, let $M_{r},N_{r}$ denote the $C^{p}$ manifolds of geodesics $x:[-r,r]\to X$, resp. $y:[-r,r]\to Y$, into which we embed $X,Y$ as before. There is no natural map between $M_{r}$ and $N_{r}$ that $\varphi$ induces. However, there is a submanifold $H_{r}\subset M_{r}$ consisting of geodesics that are perpendicular to the fibers of $\varphi$ (‘horizontal geodesics’). According to O’Neill, composition with $\varphi$ induces a map $\psi:H_{r}\to N_{r}$, see [O, Chapter 7, Lemma 45]. In the identifications

[TABLE]

$H_{r}$ corresponds to the orthogonal complement

[TABLE]

intersected with a suitable neighborhood of the zero section, and $\psi$ corresponds to the restriction of $\Phi=\varphi_{*}|H$ to this neighborhood. We see that $\psi$ is of class $C^{p}$.

Theorem 3.1**.**

If, under the above assumptions, a $G_{r}$ invariant neighborhood $M\subset M_{r}$ of $X$ admits an adapted complex structure $P$ of class $C^{p-1}$, then the neighborhood $N=\psi(M\cap H_{r})\subset N_{r}$ of $Y$ also admits an adapted complex structure of class $C^{p-1}$.

In [A] Aguilar proved a special case, when $(Y,k)$ is a quotient of $(X,h)$ by the action of a Lie group of isometries, and our proof below is partly inspired by his. We start with a general result on pushing forward involutive and adapted structures.

Lemma 3.2**.**

Let $M,N$ be manifolds of class $C^{2}$, $P\subset\mathbb{C}TM$ and $Q\subset\mathbb{C}TN$ subbundles of class $C^{1}$, and $\psi:M\to N$ a $C^{2}$ surjection such that $P=\psi_{*}^{-1}Q$.

• (a)

If $P$ is involutive, then so is $Q$.

• (b)

Suppose in addition that for some $r\in(0,\infty)$ $G_{r}$ acts on the right on both $M$ and $N$, and $\psi$ is equivariant. If $P$ is adapted to $G_{r}$, then so is $Q$.

Part (a) is the converse of [L2, Lemma 3.1].

Proof.

(a) Let $A$ be the collection of $\mathbb{C}$ valued 1-forms $\alpha$ on $N$ of class $C^{1}$ that vanish on $Q$. Involutivity means that $d\alpha|Q$ also vanish for all $\alpha\in A$. Now $P=\bigcap_{\alpha\in A}\text{Ker}\,\psi^{*}\alpha$ is known to be involutive, hence for $x\in M$

[TABLE]

i.e., $d\alpha|Q_{\psi(x)}=0$, as needed.

(b) Write $\Omega_{x}g=xg$ for $x\in M$ or $N$ and $g\in G_{r}$. That $P$ is adapted means $\Omega_{x*}S\subset P$. Since $\psi\circ\Omega_{x}=\Omega_{\psi(x)}$,

[TABLE]

which in turn means $Q$ is adapted.

Given $p\in[1,\infty]$, consider next a $C^{p+1}$ manifold $X$ with a Riemannian metric $h$ of class $C^{p}$. On the tangent bundle $\pi:TX\to X$ the tautological 1-form and the symplectic 2-form are defined by

[TABLE]

Let $(Y,k)$ be a similar Riemannian manifold with bundle projection $\varrho:TY\to Y$ and corresponding forms $\beta$ and $\sigma=d\beta$ on $TY$. If $\varphi:X\to Y$ is a submersion of class $C^{p+1}$, $\text{Ker}\,\varphi_{*}$ is a $C^{p}$ subbundle of $TX$, whose orthogonal complement is the bundle $H\to X$ of horizontal vectors.

Lemma 3.3**.**

If $\varphi:X\to Y$ is a Riemannian submersion and $\Phi=\varphi_{*}|H:H\to TY$, then $\alpha|H=\Phi^{*}\beta$ and $\omega|H=\Phi^{*}\sigma$.

Proof.

Any $u\in TX$ can be written $u=u_{0}+u_{H}$ with $u_{0}\in\text{Ker}\,\pi_{*}$ and $u_{H}\in H$. With $v\in H$ we have

[TABLE]

If $v\in H$ and $\xi\in T_{v}TX$, in light of $\varrho\circ\Phi=\varphi\circ\pi$ therefore

[TABLE]

Thus $\alpha|H=\Phi^{*}\beta$ and $d\alpha|H=\Phi^{*}d\beta$.

Proof of Theorem 3.1.

For a differentiable map $f:U\to V$ we have used, and will continue to use $f_{*}$ to denote its differential acting between $\mathbb{C}TU$ and $\mathbb{C}TV$. It will be convenient to denote the restriction $f_{*}|TU$ by $f_{*}^{\mathbb{R}}$.

The open embeddings (3.1) pull back the symplectic forms $\omega,\sigma$ to $M_{r}$ and $N_{r}$. We will keep calling the pull backs $\omega$ and $\sigma$. Since $H_{r}\subset M_{r}$ corresponds to $H\subset TX$,

[TABLE]

by Lemma 3.3. First we show that $\psi$ is a submersion and

[TABLE]

Indeed, given $x_{0}\in H_{r}$, construct through $x_{0}(0)\in X$ a submanifold $X^{\perp}$ of a neighborhood of $x_{0}(0)$ that $\varphi$ maps diffeomorphically on a neighborhood of \varphi\big{(}x_{0}(0)\big{)}\in Y. Any $y\in N_{r}$ in a suitable neighborhood of $\psi(x_{0})$ has a unique lift to a horizontal geodesic $x$ in $X$ with $x(0)\in X^{\perp}$, and the $C^{1}$ map $y\mapsto x$ is a right inverse to $\psi$ in this neighborhood; hence $\psi$ is a submersion. Further, if $\xi,\eta\in T_{x}H_{r}$, by (3.2) $\omega(\xi,\eta)=0$ is equivalent to $\sigma(\psi_{*}\xi,\psi_{*}\eta)=0$. Since $\psi_{*}T_{x}H_{r}=T_{\psi(x)}N_{r}$, (3.3) follows.

Next we recall that according to [LSz1, Corollary 5.5 and Theorem 5.6] $-\omega$ is a Kähler form on $M$ when $M$ is endowed with its adapted complex structure. Let $J:TM\to TM$ be the almost complex tensor of the adapted complex structure and let $\perp$ denote orthogonal complement in the Hermitian metric $\omega(J\xi,\eta)$ on $TM$. By (3.3)

[TABLE]

We compute

[TABLE]

whence in fact $J\text{Ker}\,\psi_{*}^{\mathbb{R}}=T(M\cap H_{r})^{\perp}$. This implies

[TABLE]

This latter bundle, that we denote $R$, is involutive since $P$ is, and endows $M\cap H_{r}$ with an involutive structure of class $C^{p-1}$ (in fact, a CR structure), that is clearly adapted to $G_{r}$. Furthermore, $\psi_{*}$ maps $E_{x}=T_{x}H_{r}\cap(\text{Ker}\,\psi_{*}^{\mathbb{R}})^{\perp}$ isomorphically on $T_{\psi(x)}N$ for $x\in M\cap H_{r}$, and so

[TABLE]

We claim that $\psi_{*}R_{x}$ depends only on $\psi(x)$ or equivalently, whether $\xi-i\eta\in R_{x}$ for $\xi,\eta\in T_{x}H_{r}$ depends only on $\psi_{*}\xi$, $\psi_{*}\eta$.

We will show this by a variant of the proof of [LSz1, Theorem 4.2]. The actions of $G_{r}$ on $M_{r},N_{r}$ induce actions on $\mathbb{C}TM_{r},\mathbb{C}TN_{r}$, that we denote $(\xi,g)\mapsto\xi g$. Various objects introduced in this proof are compatible with the $G_{r}$ actions: $\psi$ and $\psi_{*}$ are equivariant and $E\subset T(M\cap H_{r})$ is invariant. As to this latter, note that $g(t)=a+t$ acts by the geodesic flow, hence preserves $\omega$, while $g(t)=bt$ acts by scaling on $M\subset TX$, hence changes $\omega$ to a multiple. Therefore

[TABLE]

is indeed invariant.

If $\xi\in TM$, write $\xi^{\prime}=(\xi-iJ\xi)/2\in P$. Fix $x\in M\cap H_{r}$ and choose $\xi_{1},\dots,\xi_{m}\in T_{x}M$ so that $\xi^{\prime}_{1},\dots,\xi^{\prime}_{m}$, resp. $\xi^{\prime}_{1},\ldots,\xi^{\prime}_{n}$ form a basis of $P_{x}$, resp. $R_{x}$ ($m=\dim X=\text{rk}_{\mathbb{C}}\,P$, $n=\dim Y=\text{rk}_{\mathbb{C}}\,R$). In particular, $\xi_{\alpha}\in E_{x}$ for $\alpha\leq n$ are independent, and form a basis. Next choose $\eta_{1},\dots,\eta_{n}\in E_{x}$. By [LSz1, Proposition 5.1] the continuous maps

[TABLE]

are holomorphic over int$\,G_{r}$. Hence for $g\in\text{int}\,G_{r}$, except perhaps a discrete subset $\Delta\subset\text{int}\,G_{r}$, the vectors $(\xi_{\alpha}g)^{\prime}$ are linearly independent. Therefore there are meromorphic functions $F_{\alpha\beta}$ on $\text{int}\,G_{r}$ such that

[TABLE]

Moreover, as the $(\xi_{\beta}g)^{\prime}$ are independent at $g=e$ as well, $F_{\alpha\beta}$ have a limit at $e$, that we denote $F_{\alpha\beta}(e)$. Let $\Gamma\subset G_{r}$ consist of constant maps $t\mapsto a$, where $|a|<r$. Since $N_{r}g=Y$ when $g\in\Gamma$,

[TABLE]

In addition, $\xi_{\alpha}g,\eta_{\alpha}g\in E_{xg}$, and $\xi_{\alpha}g$ for $\alpha=1,\dots,n$ are linearly independent when $g\in\Gamma\setminus\Delta$ (because the $(\xi_{\alpha}g)^{\prime}$ are). Therefore there are unique functions $f_{\alpha\beta}:\Gamma\setminus\Delta\to\mathbb{R}$, $\alpha,\beta\leq n$, such that

[TABLE]

when $g\in\Gamma\setminus\Delta$. It follows that $(\eta_{\alpha}g)^{\prime}=\sum_{\beta=1}^{n}f_{\alpha\beta}(g)(\xi_{\beta}g)^{\prime}$, and by (3.6) that $F_{\alpha\beta}=f_{\alpha\beta}$ on $\Gamma\setminus\Delta$ for $\alpha,\beta\leq n$. Thus the meromorphic functions $F_{\alpha\beta}$ for $\alpha,\beta\leq n$ are uniquely determined by $f_{\alpha\beta}$ which, in view of (3.7), are determined by $\psi_{*}\xi_{\alpha},\psi_{*}\eta_{\alpha}$. But $\xi_{\alpha}+i\eta_{\alpha}\in R_{x}$ if and only if $\eta^{\prime}_{\alpha}=i\xi^{\prime}_{\alpha}$ ($\alpha\leq n)$. Since $\eta^{\prime}_{1},\dots,\eta^{\prime}_{n}\in R_{x}$ and $\xi^{\prime}_{1},\ldots,\xi^{\prime}_{n}$ is a basis, (3.6) implies

[TABLE]

Hence $\xi_{\alpha}+i\eta_{\alpha}\in R_{x}$ for $\alpha\leq n$ if and only if $F_{\alpha\beta}(e)=i\delta_{\alpha\beta}$ ($\alpha,\beta\leq n$), and depends only on $\psi_{*}\xi_{\alpha}$, $\psi_{*}\eta_{\alpha}$, and their orbit under $G_{r}$.

Therefore $\psi_{*}R_{x}=Q_{\psi(x)}$ indeed depends only on $\psi(x)$ and defines a $C^{p-1}$ subbundle $Q\subset\mathbb{C}TN$, which, by (3.5) and by Lemma 3.2 is an adapted complex structure on $N$.

4 Riemannian submersions and analyticity

Finally we prove the following version of Theorem 1.1.

Theorem 4.1**.**

Let $p\in(3,\infty]$ be nonintegral, $Y$ a $C^{p+1}$ manifold endowed with a $C^{p}$ Riemannian metric $k$, and $(X,h)$ an analytic Riemannian manifold. If $\varphi:X\to Y$ is a surjective Riemannian submersion of class $C^{p+1}$, then $Y$ admits an analytic structure in which $h$ is analytic. Moreover, this analytic structure and the original $C^{p+1}$ manifold structure of $Y$ share the same underlying $C^{p}$ manifold structure.

Proof.

Fix $r\in(0,\infty)$. Let $M_{r},N_{r}$ be the space of geodesics $[-r,r]\to X$, resp. $Y$, into which $X$ and $Y$ are embeded as the space of constant geodesics. Possibly after shrinking $X$ and $Y$, some neighborhood $M$ of $X\subset M_{r}$ will admit an analytic adapted complex structure $P$ (Theorem 2.4). By Theorem 3.1 $P$ induces an adapted complex structure $Q$ of class $C^{p-1}$ on a neighborhood $N\subset N_{r}$ of $Y$. The conclusion of the theorem now follows from Theorem 2.5.

It is a natural question what happens if we weaken the assumption of Theorem 4.1 on $(X,h)$ to $C^{\infty}$ smoothness. Does it imply that $(Y,k)$ must be $C^{\infty}$? We do not know the answer.