# Tagged Questions

Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].

79 views

### Nash-type theorems for Poisson manifolds

My question comes as a natural follow-up of the previous one which concerned symplectic manifolds: if $(M, P)$ is a Poisson manifold, what embedding theorems are there into some target space (I am ...
414 views

### Is there a Nash-type theorem for symplectic manifolds?

If $(M, \omega)$ is a symplectic manifold, is it possible to embed (or injectively immerse) it symplectically into a sufficiently large $(\Bbb R ^{2N}, \Omega)$, (with the usual symplectic structure)? ...
111 views

### Existence of non-negative extensions of smooth functions on axes

I am struggling to solve an extension problem of smooth functions, and I would like someone to help me. The setting is as follows: Let $X_1$, $X_2$, and $T$ be either the real lines $\mathbb R$ or ...
283 views

### Recovering a smooth manifold from its tensor fields

1) Consider the algebra $\mathcal T (M) = \bigoplus \limits _{p, q \ge 0} \mathcal T ^{p, q} (M)$, where $\mathcal T ^{p, q} (M)$ is the space of tensor fields of type $(p,q)$. I should endow it with ...
57 views

### material derivative and relation to Riemannian metric

For each $n$ let $N_t$ be an embedded smooth hypersurface in $\mathbb{R}^n$ of dimension $n-1$. $\{N_t\}_t$ is a family of hypersurface that is evolving with some velocity $V$. Smooth functions on $N$...
148 views

159 views

235 views

### global section of affine $C^\infty$-scheme

I'm reading Algebraic Geometry over $C^\infty$-rings. It is written that "If $\mathfrak{C}$ is not finitely generated then $\Phi_{\mathfrak{C}}:\mathfrak{C}\rightarrow \Gamma(\text{Spec}\mathfrak{C})$...
42 views

### Hilbert Manifold structure on global sections of a sheaf

Let $<U_i,\phi_i>$ be an atlas on a Hilbert manifold $M$. If $F(M,U_i)$ is a Hilbert space for every $U_i$ then does this imply that $F(M,M)$ can be made into a Hilbert manifold also with ...
172 views

### Unique Kahler-Einstein metric $g$ with $\mathrm{Ricc}(g)=-g$ when first Chern class $C_1(M)<0$: $\mathrm{Ricc}(h)=-g\,\Rightarrow\,h=cg$ for $c>0$?

On a compact Kahler manifold, let $g$ be the unique Kahler-Einstein metric with $\mathrm{Ricc}(g)=-g$, proved to exist by Yau and Aubin when the first Chern class $C_1(M)<0$. Question: Does $g$ ...
231 views

### Does every smooth manifold carry a gaussian random field?

Let $M$ be an arbitrary finite-dimensional smooth manifold. For simplicity, let's assume that $M$ has no boundary. Does there always exist a gaussian random field with constant variance on $M$? If not,...
177 views

### A tangential fixed point property for manifolds embedded in Euclidean spaces

Assume that $M$ is a compact orientable manifold which is embedded in some Euclidean space $\mathbb{R}^{N}$ We say that $M\subset \mathbb{R}^{N}$ has the tangential fixed point property if for ...
129 views

280 views

### Question about theorem in Arnold's book on action-angles variables

I have a question about the action-angle theorem on p. 283 in Arnold's textbook on classical mechanics.(I added the link to this book in the last part of this question) If you don't have the book or ...
221 views

### Homotopies with prescribed regular values

Let $M_1$ and $M_2$ be connected smooth manifolds and let $f_0,f_1:M_1 \rightarrow M_2$ be homotopic smooth maps such that some fixed point $p \in M_2$ is a regular value for both $f_0$ and $f_1$. ...
101 views

### Retract embedding of $S^{n}$ in its unit tangent bundle

Acording to the comment of Mark Grant and the answer of Ryan Budney, I revise the question: For what even $n$, there is a retract embedding of of $S^n$ in its unit tangent bundle?
237 views

### Hodge de Rham operator and orientability

Let $(M,g)$ be a Riemannian manifold. One can consider the exterior algebra bundle $\Lambda(T^*M)$. The sections of this bundle are differential forms, to be noted by $\Omega^k(M)$. One can consider ...
530 views

### characteristic classes of homotopy equivalent manifolds

Let $M,N$ be two manifolds of different dimensions. Suppose $M\simeq N$, i.e. $M$ is homotopy equivalent to $N$. Do the Stiefel-Whitney classes of the tangent bundles of $M$ and $N$ equal  w(TM)=w(...
91 views

A density matrix is a matrix $\rho \in \mathscr{D}:=\{A \in \mathbb{C}^{n \times n}; A^*=A; \operatorname{tr}(A)=1; A \ge 0\}.$ In Quantum Mechanics it is natural to look at a group action $\Phi: U(... 2answers 131 views ### Does an analytic tensorial Lie structure on$S^2$gives a fiberwise Abelian Lie algebra structure? Motivated by the answer to this question we ask: Is it true to say that for every real analytic tensorial Lie algebra structure$\alpha$on$\chi^{\infty}(S^2)$, all fibers are necessarily ... 1answer 128 views ### Manifold_Lie algebra compatibility In this question we try to improve some parts of this post as follows: What is an example of a manifold$M$and a Lie algebra$L$(with the same dimension) such that$M$does not admit ... 1answer 115 views ### Existence of tubular neighborohoods of locally flat topological embeddings Suppose$X$is a topological manifold and$Y \subset X$is a locally flat submanifold. We know that$Y$doesn't necessarily have a tubular neighborhood. My definition of a tubular neighborhood of$Y$... 2answers 417 views ### Riemannian metrics preserved by diffeomorphisms Let$f \neq Id$be a diffeomorphism (of a smooth manifold$M$) which admits some Riemannain metric on$M$making it an isometry. How many different metrics are preserved by$f$? Note that$Met(f)=\{g|...
What is an example of a closed 4-manifold $M$ such that $M$ is parallelizable and $M$ is topologically (or at least smoothly) irreducible? Topological irreducible: it is not homemorphic to ...