The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in this current post, we are highly reducing and ignoring the requirement "Geodesibility"

Let $X$ be a nonvanishing vector field on a Riemannian surface $(M,g)$. For $q\in M$, the geodesic curvature of the orbit of$ X$ passiing $q$ is denoted by $\kappa_g(q)$.

**Definition:** A non vanishing vector field $X$ on a $2$-manifold $M$ satisfies WG property (weak geodesible) if there is a Riemannian metric $g$ on $M$ such that the smooth function $\kappa_g(q).|X(q)|$ belongs to the range of the derivation operator $X(u)=X.u=du(X)$. Namely there exist a smooth function $u$ on $M$ with $X.u=\kappa_g |X|$.

**Remark 1:** Obviousely every geodesible flow is a WG flow.

**Question 1:** Is the Vander Pol vector field $(V)$ bellow a WG vector field on $\mathbb{R}^2\setminus \{0\}$?$$(V)\;\;\;\begin{cases} x'=y-(x^3-x)\\y'=-x\end{cases}$$

**Question 2:** Is there a negative curvature Riemannian metric $g$ on the punctured plane for which $\kappa_g |V|$ lies in the range of derivation associated to $(V)$.

**Proposition:** If the answer to the second question is affirmative, then we obviousely have an alternative proof for the fact that $(V)$ has at most one limit cycle.

**Proof:** If $\gamma_1, \gamma_2$ are two limit cycles then $\int_{\gamma_i} \kappa_gds=\int_{\gamma_i} \kappa_g|V|dt=0$. Applying the Gauss Bonnet theorem to the annular Region bounded by $\gamma_1, \gamma_2$ gives us a contradiction.

**Remark 2:** As we mentioned in the head of this post, we are trying to find some resolutions to difficulties appeared in the following efforts for consideration of Limit cycles as closed geodesics.
So in this more flexible approach we do not require that all orbits would be geodesics.(A very rigid and non generic situation). On the other hand we are trying to find a relation for the problem of investigation of the range of the differential operator associated to a vector field.

Let $C = \{ e _{1}, \cdots e _{n} \}$, where each $e _{i}$ are unit vectors in $\ell ^{p, \infty}$, and $1 < p < \infty$. I want prove that $\overline{conv}(C)$ is diametral. My doubt is: $\Vert x - y \Vert _{p, \infty} \leqslant 1$ for all $x, y \in \overline{conv}(C)$, where \begin{align*} \Vert x \Vert _{p, \infty} = \max \{ \Vert x ^{+}\Vert _{p}, \Vert x ^{-} \Vert _{p} \} \end{align*} and \begin{align*} (x ^{+}) ^{i} = \max \{ x _{i}, 0 \} = \frac{\vert x _{i}\vert + x _{i}}{2} \quad \mbox{and} \quad (x ^{-}) ^{i} = \max \{ -x _{i}, 0 \} = \frac{\vert x _{i}\vert - x _{i}}{2} \end{align*} for all $x _{i} \in \ell ^{p, \infty}$?.

I would like a result of the following form:

For every algebraic curve $C$ in $\mathbb{R}\mathbf{P}^{n-1}$, there exists an explicit and easy-to-compute $\epsilon=\epsilon(C)>0$ such that points $x,y\in C$ satisfying $\mathrm{dist}(x,y)<\epsilon$ necessarily reside in a common connected component of $C$.

For example, $\epsilon$ might be a function of the "complexity" of the polynomials that define $C$ (e.g., degrees, size of coefficients, etc.). Is such a result available in the current literature?

The classical bang-bang theorem is usually stated for linear systems (e.g. Control Theory from the Geometric Viewpoint by Agrachev-Sachkov, p. 209). Sussman proved a nice generalization for systems affine in control $\dot{x}=f_0(x)+f_1(x)u$, which gives geometric conditions in terms of Lie brackets of $f_i$ for the control to be bang-bang (or not), bounds on the number of switches, etc. They depend only on the Lie structure (which is the analog of the Riemannian curvature here), and hence are invariant under coordinate transformations, linear or nonlinear. Unfortunately, the versions in his papers (e.g. An introduction to the coordinate-free Maximum Principle, p.65) are always stated for the time minimization problem only. Wikipedia mentions, without a reference, that ``bang-bang solutions also arise when the Hamiltonian is linear in the control variable". Examples readily confirm that for functionals of the form $\int_0^T L_0(x)+L_1(x)u\,dt+l(x(T))$ (the so-called Bolza problem).

But I could not find the theory spelled out for such problems, e.g. general conditions that rule out singular controls, bounds on the number of switches, etc., in terms of $f_i,L_j$. I am aware of the formal trick that eliminates the integral by introducing new "time", but it involves division of $f_i$ by $L$, and even if $L_1=0$ tracking formulas, and technical assumptions, through the transformation is messy.

Is there some principled difficulty with extending the theory to the Bolza functionals? Is it written up somewhere explicitly?

It is a normal practice to use a minimal set of operators in logical systems and construe the other operators as abbreviations.

Let's look at the propositional logic:

If $\mathcal{V}$ denotes the set of variables $v_0, v_1, \dots$ we can define the set of propositional formulas as the smallest set $L_0 \supseteq \mathcal{V}$ of strings closed under building the negation and the conjunction. So we have: $$L_0 = \mathcal{V} \cup \{\neg \varphi \,;\, \varphi \in L_0\} \cup \{(\varphi \wedge \psi) \,;\, \varphi,\psi \in L_0\}$$ Then we introduce $(\varphi \vee \psi)$, $(\varphi \rightarrow \psi)$ and $(\varphi \leftrightarrow \psi)$ iteratively as abbreviations for $\neg(\neg \varphi \wedge \neg \psi)$, $(\neg \varphi \vee \psi)$ and $((\varphi \rightarrow \psi) \wedge (\psi \rightarrow \varphi))$, respectively. We can be more exact and define an extended set of formulas $L$ such that $$L = \mathcal{V} \cup \{\neg \varphi \,;\, \varphi \in L\} \cup \{(\varphi \ast \psi) \,;\, \ast \in \{ \wedge,\vee,\rightarrow, \leftrightarrow \}, \varphi,\psi \in L\}$$ and define the "one-step-reduction" $R : L \to L$ recursively by $$\alpha \mapsto \begin{cases} \alpha , & \alpha \in \mathcal{V} \\ \neg R(\varphi), & \alpha = \neg\varphi \\ (R(\varphi)\wedge R(\psi) ), & \alpha = (\varphi \wedge \psi) \\ \neg(\neg \varphi \wedge \neg \psi), & \alpha = (\varphi \vee \psi) \\ (\neg \varphi \vee \psi), & \alpha = (\varphi \rightarrow \psi) \\ ((\varphi \rightarrow \psi) \wedge (\psi \rightarrow \varphi)), & \alpha = (\varphi \leftrightarrow \psi) \\ \end{cases}$$

**Proposition 0:**
For all $\alpha \in L$ exists an $n \in \mathbb{N}$ such that
$$R^n(\alpha) \in L_0$$
Therefore we can define a "reduction" $Red : L \to L_0$ by
$$\alpha \mapsto R^d(\alpha), \quad\text{where } d := \min \{n \in \mathbb{N} \,;\, R^n(\alpha) \in L_0\}$$

In my diploma thesis I needed a more general result:

Let $\langle a_1, \dots, a_n \rangle$ denote the $n$-tuple of $a_1, \dots, a_n$. We define $\mathcal{V} := \{v_k \,;\, k \in \mathbb{N}\}$, where $v_k := \langle 0, k \rangle$ (solely to be sure that variables are distinct from the other formulas I'll define).

Choose a set $I \subseteq \mathbb{N}\setminus \{0\}$ of "operators". We assign an arity $n_i \in \mathbb{N}$ to each operator $i \in I$. Now we can define a set of formulas $L$ such that: $$L = \mathcal{V} \cup \{\langle i, \vec{\varphi} \rangle \,;\, i \in I, \vec{\varphi} \in L^{n_i}\}$$ So the "application" of an operator $i \in I$ to formulas $\varphi_1, \dots, \varphi_{n_i}$ is given by $\langle i, \langle \varphi_1, \dots, \varphi_{n_i} \rangle \rangle$. The "operators used in a formula" are determined by $$ \operatorname{op} : L \to \mathcal{P}(I), \alpha \mapsto \begin{cases} \emptyset, & \alpha \in \mathcal{V} \\ \{i\} \cup \bigcup_{1 \le k \le n_i} \operatorname{op}(\varphi_k), & \alpha = \langle i, \langle \varphi_1, \dots, \varphi_{n_i} \rangle \rangle \end{cases}$$

Some of the operators $I_0 \subseteq I$ we call primary operators. The set of "primary" formulas $L_0$ is determined by: $$L_0 = \mathcal{V} \cup \{\langle i, \vec{\varphi} \rangle \,;\, i \in I_0, \vec{\varphi} \in L_0^{n_i}\}$$

For each non-primary operator $i \in I \setminus I_0$ choose a "defining formula" $\delta_i$ (that contains exactly the variables $v_1, \dots , v_{n_i}$) such that the relation ${\prec} \subseteq I \times I$ given by $$ i \prec j \quad\text{:iff}\quad j \in I \setminus I_0 \text{ and } i \in \operatorname{op}(\delta_j) $$ is well-founded. (This prevents circular "operator definitions".)

If we define for $\vec{\psi} = \langle \psi_1, \dots, \psi_{n} \rangle \in L^n$ a substitution $\operatorname{Sub} : L \to L, \alpha \mapsto \alpha[\vec{\psi}]$ by $$ \alpha[\vec{\psi}] = \begin{cases} \psi_k, & \alpha = v_k, \ k \in \{1,\dots,n\} \\ \alpha, & \alpha = v_k, \ k \notin \{1,\dots,n\} \\ \langle i, \langle \varphi_1[\vec{\psi}], \dots, \varphi_{n_i}[\vec{\psi}] \rangle \rangle, & \alpha = \langle i, \langle \varphi_1, \dots, \varphi_{n_i} \rangle \rangle \\ \end{cases} $$ we can "unpack" an $\langle i, \vec{\varphi} \rangle$ with $i \in I \setminus I_0$ to $\delta_i[\vec{\varphi}]$. More precisely: We define the "one-step-reduction" $R : L \to L$ by $$\alpha \mapsto \begin{cases} \alpha , & \alpha \in \mathcal{V} \\ \langle i, \langle R(\varphi_1), \dots, R(\varphi_{n_i}) \rangle \rangle , & \alpha = \langle i, \langle \varphi_1, \dots, \varphi_{n_i} \rangle \rangle, i \in I_0 \\ \delta_i[\vec{\varphi}] , & \alpha = \langle i, \vec{\varphi} \rangle, i \notin I_0 \\ \end{cases}$$

Then we get the analogue of Proposition 0.

**Proposition 1:**
For all $\alpha \in L$ exists an $n \in \mathbb{N}$ such that
$$R^n(\alpha) \in L_0$$
Therefore we can define a "reduction" $Red : L \to L_0$ by
$$\alpha \mapsto R^d(\alpha), \quad\text{where } d := \min \{n \in \mathbb{N} \,;\, R^n(\alpha) \in L_0\}$$

Intuitively it is almost obvious that Proposition 1 (and 0) is true. But the proof of it in my thesis is quite complicated (I defined 6 other well-founded relations including a relation on the Kleene closure of the Kleene closure of $I$). My **question** is therefore:

**Does someone have an idea for a simpler proof or is there a common theorem which I can use here?**

Remark:

To see that Proposition 1 is a generalisation of Proposition 0:

- Let $I = \{i_\neg, i_\wedge, i_\vee, i_\rightarrow, i_\leftrightarrow\}$ be an arbitrary subset of $\mathbb{N}\setminus \{0\}$ with 5 elements
- $I_0 := \{i_\neg, i_\wedge\}$
- $n_{i_\neg} := 1$ and $n_i := 2$ for $i \in I \setminus \{i_\neg\}$
- write $\neg \varphi$ for $\langle i_\neg, \langle \varphi \rangle \rangle$ and $(\varphi \ast \psi)$ for $\langle i_\ast, \langle \varphi, \psi \rangle \rangle$ if $\ast \in \{\wedge, \vee, \rightarrow, \leftrightarrow\}$
- define $\delta_{i_\vee} := \neg (\neg v_1 \wedge \neg v_2)$, $\delta_{i_\rightarrow} := (\neg v_1 \vee v_2)$ and $\delta_{i_\leftrightarrow} := ( (v_1 \rightarrow v_2) \wedge (v_2 \rightarrow v_1) )$

Let $\epsilon_n$ be a sequence in $\{-1,1\}^{\mathbb Z_+}$. For simplicity, assume that $\epsilon_n$ is just the Thue-Morse sequence with symbols $1$ and $-1$ (although the following definition is supposed to make sense for more general $\epsilon_n$ too)

We define partial Fourier amplitudes

$$G_N(q)=\sum_{1\leq n\leq N} \epsilon_n \exp(inq)$$

and the corresponding intensities

$$S_N(q)=\frac{1}{N}|G_N(q)|^2$$

Does the limit

$$d\mu(q)=\lim_{N\to\infty} \left[ S_N(q) \frac{dq}{2\pi}\right]$$

make sense? How do we prove rigorously that it exists?

For context, this definition of $d\mu(q)$ (the Fourier intensity measure) comes from (2.3) in the following paper:

Luck, J. M. "Cantor spectra and scaling of gap widths in deterministic aperiodic systems." Physical Review B 39, no. 9 (1989): 5834.

This paper can be found here (paywalled)

I asked this question previously on Math Stackexchange here. But I think it might be a better fit for Math Overflow

Let $J_0(p)$ be Jacobian of the modular curve $X_0(p)$ over $\mathbb Q$ where $p$ is a prime, consider the subring $\mathbb T$ inside $End_{\mathbb Q}(J_0(p))$ generated by Hecke operators $T_n$ for all $(n, p)=1$. Then what are $\mathbb T$ and $End_{\mathbb Q}(J_0(p))$ as $\mathbb Z$-algebras? Do we have $T=End_{\mathbb Q}(J_0(p))$? If not, what is the index /difference?

If we tensor both sides with $\mathbb Q$, then they are equal by the work of Ribet, and the ring structure is given by products of totally real fields which are precisely coefficient fields of weight $2$ level $p$ cusp forms. Here I am more interested in the integral structures.

Let $M$ be an $S_n$-lattice (so it is free as an abelian group), and assume that $M$ is projective (i.e. direct summand of some $\mathbb Z[S_n]^m$). A theorem of Swan implies that $M$ is stably permutation, that is, $M\oplus \mathbb Z[E]=\mathbb Z[F]$, for some finite $G$-sets $E$ and $F$.

The theorems that go into the proof of this result are quite abstract and not constructive, I think. Is there a way to control $E$ and $F$? For example, can I always choose them so that no orbit has an $A_n$-stabilizer? Or such that every stabilizer is a $p$-group?

Let $\mathfrak{g}$ be a real semisimple Lie algebra. We know that a coadjoint orbit $\mathcal{O} \hookrightarrow \mathfrak{g}^*$ carries a natural symplectic form $\omega$, namely the Kirillov-Kostant-Souriau form. Is there something tying $\omega$ to the Killing form $\kappa$ of $\mathfrak{g}$? It seems to me like there should be something like that, since both objects are intrinsic to $\mathfrak{g}$. I'm thinking something along the lines of, "if $\mathcal{O}$ is a semi-Riemannian submanifold of $\mathfrak{g}^*$, then the Levi-Civita connection of $\kappa$ (defined on the dual $\mathfrak{g}^*$ in the obvious way) induces a symplectic connection on $\mathcal{O}$", or something.

Let $Sing:C \rightarrow SSet$ be the functor sending a CW complex to its singular complex (a simplicial set).

Let $∣-∣: SSet \rightarrow C$ be the geometric realization functor.

For every $X$ we have a natural map that is a homotopy equivalence $\epsilon_X: ∣Sing(X)∣\rightarrow X$.

Is there a natural homotopy inverse? Said differently, is there a natural transformation $\text {Id} \Rightarrow∣Sing(-)∣$ consisting of homotopy inverses to the $\epsilon_X$’s?

Assume we want to find a real random variable $X$ s.t. the logarithm of $|X|$ is distributed as $X$ itself, $$X \sim \log |X| \;.$$ The motivation for this could be to find a distribution to sample from the real line, in a way that $X$ is scale-invariant, and the scale of $X$ is scale-invariant, and so on. While that sounds very ill-behaved, we can take the equality to the probability distribution function (say for $x > 0$), $$f(x) \, dx = f(\log x) \, d \log x = \frac{f(\log x)}{x} \, dx \;.$$ There is hence a functional equation $$f(\log x) = x \, f(x)$$ for the distribution function of this strange $X$.

**Q:** *Can this functional equation be solved?*

I do not know how to attack such a problem, but one observation is readily made. Insert the tetration ${}^n e$, i.e. ${}^0e = 1$, ${}^1e = e$, ${}^2e = e^e$, ${}^3e = e^{e^e}$, etc. into the functional equation, and you get $f(0) = f(1)$, $f(1) = e \, f(e)$, $f(e) = e^e \, f(e^e)$, $f(e^e) = e^{e^e} \, f(e^{e^e})$, etc., or $$f({}^n e) = {}^{n+1} e \, f(^{n+1} e) \;.$$ The recursion is easily solved, $$f({}^n e) = \frac{f(0)}{\prod_{k=0}^{n} {}^k e}\;,$$ and from the few numbers that don't overflow, we see that while the ordinate on the right falls off rapidly, the abscissa on the left grows rapidly, leading to an overall quite well-behaved distribution (in the three points that we know, with $f(0) = 1$):

This fact is what made me curious about this distribution. I have two more questions:

**Q:** *Is there a closed-form expression for the product of tetrations?*

**Q:** *Is there a natural way to extend the iteration result for real values?*

I would also appreciate tag suggestions.

The comment below points to a further idea. If the distribution is taken to be symmetric about the origin, $f(-x) = f(x)$, then $$f(1/x) = x \, f(-\log x) = x \, f(\log x) = x^2 f(x) \;,$$ which suggests that $f(x)$ grows rapidly as it approaches $0$, $$f(({}^n e)^{-1}) = ({}^n e)^2 \, f({}^n e) = \frac{({}^n e)^2 \, f(0)}{\prod_{k=0}^{n} {}^k e} \;.$$ It also yields a couple more points to plot:

Let $\pi:\mathcal{X} \to S$ be a flat family of affine curves contained in $\mathbb{C}^n$ for $n \ge 3$ i.e., $\mathcal{X} \hookrightarrow \mathbb{C}^n_S$ and the inclusion commutes with the natural morphism to $S$. Let $p \in \mathbb{C}^n$ be a general point (not intersecting any $\mathcal{X}_s$) and $\phi:\mathbb{C}^n_S \backslash (p \times S) \to \mathbb{C}^{n-1}_S$ be the trivial deformation of the linear projection from the point $p$. Let $\mathcal{Y} \subset \mathbb{C}^{n-1}_S$ be the image of $\mathcal{X}$ under the morphism $\phi$. Is $\mathcal{Y}$ going to be $S$-flat (under the natural morphism to $S$)?

Is there an example of $G$, $\rho$ as below?

$G$ is a locally compact group.

$\rho$ is an irreducible continuous representation of $G$ on a complex Hilbert space $V$. This means that we have a continuous homomorphism from $G$ to the group of bounded linear operators on $V$ with bounded inverse, such that $G \times V \rightarrow V$ is continuous. And there are no nontrivial closed invariant subspaces.

Schur's lemma fails for $\rho$.

(Asked first on MSE: https://math.stackexchange.com/questions/3096704/failure-of-schurs-lemma-for-topological-group-representations)

**Added later:**
For concreteness, I will record below one version of Schur's lemma that I have in mind. (But I'm not too picky about this.)

*Schur's lemma:* Every bounded operator commuting with $\rho(G)$ is a scalar.

Let $X = Y = \mathbb{R}^d$ and let $\nu$ be a probability measure on $\mathbb{R}^d$. Consider the collection of probability measure $\pi$ on $X\times Y$ such that $\pi$ has $y$-marginal $\nu$:

$$ \Pi(\nu) = \{\pi: \pi(X,dy) = \nu(dy)\}. $$

Let $f:X\times Y \mapsto \mathbb{R}$ be a measurable function (you can assume it is continuous and has nice growth condition) such that the partial minimization

$$ \phi(y) = \inf\{f(x,y):x\in X\} $$ is measurable. My question: is it true that

$$ \inf_{\pi \in \Pi(\nu)} \int f(x,y) d\pi = \int \phi(y) d\nu, $$ and if so, can it be proved without using any sort of measurable selection?

Topological automorphic forms (TAF) were introduced by Mark Behrens and Tyler Lawson in 2007, being to Shimura varieties what topological modular forms (TMF) is to the moduli stack of elliptic curves.

While a previous question asked about TMF, there does not seem to be a similar question for TAF (and indeed not many questions about it at MathOverflow at all).

**Question:** How do TAF fit into homotopy theory, what are the connections between them and other concepts, such as TMF, and why should a homotopy theorist care about them?

For instance, the nLab page on TAF has the following comparison between $\mathrm{KO}$, $\mathrm{TMF}$, and $\mathrm{TAF}$: (Reproduced here only partially.)

(The title of this question is inspired by this one, and it is * not*, in any possible way, meant as depreciative.)

I was reading the following question: A stochastic process that is 1st and 2nd order (strictly) stationary, but not 3rd order stationary

The following matrix is given representing the correlation structure of a stochastic process $X[t], X[t+1], X[t+2]$:

$$\left[\begin{array}{cc} \sigma^2 & a & b \newline a & \sigma^2 & c \newline b & c& \sigma^2 \end{array}\right]$$

I noticed that the correlation structure matrix is not a Toeplitz matrix. However, this seems to result in a contradiction: if you take $t=k$ then you find that the correlation between $X(k+1)$ and $X(k+2)$ is $c$, whereas if you take $t=k+1$, then you find that the correlation between $X(k+1)$ and $X(k+2)$ is $a$. This seems to mean that the matrix is not a valid correlation structure unless it is a Toeplitz matrix.

Is my reasoning valid?

I understand this is likely to be closed as off-topic, but I thought I would give it a try regardless.

I'm curious if there are any good math books out there that take a "casual approach" to higher level topics. I'm very interested in advanced math, but have lost the time as of late to study textbooks rigorously, and I find them too dense to parse casually.

By "casual", I mean something that goes over maybe the history of a certain field and its implications in math and society, going over how it grew and what important contributions occurred at different points. Perhaps even going over the abstract meaning of famous results in the field, or high level overviews of proofs and their innovations. Conversations between mathematicians at the time, stories about how proofs came to be, etc.

I suppose the best place to start would be math history books, but I was curious what else there may be. Are there any good books out there like this? What are your recommendations?

We have the result that $ZFCfin$, the usual $ZFC$ axioms with the axiom of infinity replaced by its negation, is bi-interpretable with $PA$, first order Peano Arithmetic. We also know of a natural way to weaken $PA$ into fragments, by restricting the induction axiom schema to forumlae of a specific complexity, so $Q+I\Sigma_3$ is the non inductive axioms of $PA$ plus induction restricted to formulae of at most $\Sigma_3^0$ complexity in the language of first order arithmetic.

Does weakening the axiom of separation and the axiom of replacement in $ZFCfin$ result in the above outlined fragments of PA? For example, does weakening the two axiom schema to formulae of $\Sigma_3$ complexity in the language of set theory give a set theory bi-interpretable with $Q+I\Sigma_3$? Or does the axiom of powerset also need to be dropped and then replacement replaced with collection?*

If restricting separation and replacement is the correct way of getting bi-interpretable theories, does a theory bi-interpretable with $Q$, Robinson Arithmetic, result from dropping both axiom schemes entirely?

*The reason I say this is because I know that $KP$, Kripke-Platek set theory, does away with powerset and without powerset collection and replacement are not equivalent. I am wondering if that plays a factor and if it isn't too much for one question, if anyone can explain the interplay between getting rid of powerset and the strength of fragments of arithmetic.

A warped product metric on the "cone" $\tilde{M} = \mathbb{R}^{+} \times M$ is $\tilde{g} =dr^2 + r^2g_M$ where $g_M$ is the metric on $M$.

If we know the holonomy group of the manifold $(M,g_M)$, what can we say about the holonomy group of $(\tilde{M}, \tilde{g})$?

Any references?

Edit: Here is a much more specific question:

In this paper: https://arxiv.org/pdf/math/0703231.pdf, lemma 2 states that the $\tilde{g}$ is Ricci-flat if and only if $g_M$ is Einstein with Einstein constant $(n-1)$. So in the case where $M$ is a $G_2$ manifold, it has Einstein constant 0 (on account of being Ricci flat). So that means the cone metric over a $G_2$ manifold is never Ricci-flat (and hence, for example, can't have holonomy Spin(7)).

Is that correct?

Assume M is a closed manifold. Does the set of smooth foliations on M form a Frechet submanifold inside the Frechet manifold consisting of smooth distributions?

If not, can the set of smooth foliations(which is a closed set in the space of smooth distributions) atleast be expressed as a retract of some open set?