1
vote
0answers
50 views

Cover a set with balls centered at smooth functions (Ascoli theorem)

Assume $M$ to be a compact $n$-dimensional manifold, endowed with a complete metric. Let us consider the space $C^\infty(M)$ endowed with the standard $C^\infty$ topology, i.e. generated by the ...
3
votes
0answers
167 views

Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...
4
votes
0answers
145 views

Atlas of a manifold as a Sheaf

--Hopefully this question does not dublicate another-- In this question Tom Goodwillie pointed out, that the 'atlas part' of the definition of a smooth manifold can be redefined in terms of sheaves. ...
12
votes
2answers
509 views

Does every compact manifold exhibit an almost global chart

Let $M$ be a compact connected manifold. Is there a chart $\Psi:U \to \mathbb{R}^n$ such that the closure of $U$ is $M$? This is true for $S^n, T^n, K$, all compact surfaces, etc. If it is not true in ...
20
votes
1answer
516 views

Ellipses on spheres (and other surfaces)

Define an ellipse $E$ on a sphere as the locus of points whose sum of shortest geodesic distances to two foci $p_1$ and $p_2$ is a constant $d$. There are conditions on $\{ p_1, p_2, d \}$ for this ...
24
votes
8answers
3k views

Why differential forms are important?

Importance of differential forms is obvious to any geometer and some analysts dealing with manifolds, partly because so many results in modern geometry and related areas cannot even be formulated ...
2
votes
0answers
63 views

Strictly Convex Smoothing of a function defined on an affine manifold

A function defined on an interval is strongly convex if it has positive definite second derivative. A function on an affine manifold is strongly convex if its restriction to each line is. An affine ...
3
votes
2answers
233 views

The group of diffeomorphisms with compact support

Let $M$ be a topological/differentiable manifold. Is there any topology on the group of homeomorphisms/diffeomorphisms with compact support, turning it into a (locally-)compact topological group? (My ...
1
vote
1answer
369 views

A multilinear question and its smooth version

Let $E$ and $F$ be two finite dimensional vector spaces. For every $k\in \mathbb{N}$, $E^{k}$ has a natural vector space structure and is isomorphic to $E\otimes \mathbb{R}^{k}$, in a natural way. ...
1
vote
1answer
202 views

A special type of transitivity

Let $M$ be a smooth orientable manifold with volume form $\Omega$. Fix two pints $x,y \in M$. Put $A$=all volume preserving diffeomorphism of M which maps $x$ to $y$. Define $B$=All linear volume ...
3
votes
0answers
146 views

smoothing a current

Let $M$ be a smooth oriented manifold of dimension $n$ and $T$ a current of dimension $k$ on $M$. Let $\phi:P\times M \to M$ be a proper smooth family of diffeomorphisms of $M$ (i.e. $P$ is a smooth ...
1
vote
0answers
85 views

Riemann normal coordiantes and change of metric

Le $(M,g)$ be Riemannian manifold. Fix point $p\in M$. We can define the map $$\exp: U \subset T_p M \rightarrow M$$ $$\exp(X) = \gamma_{p,X}(1)$$ where $t\mapsto \gamma_{p,X}(t)$ is geodesic such ...
4
votes
1answer
188 views

Exercises around Diffeological Spaces or a Diffeologic Atlas Theory

Is there a Book or a bunch of exercises to get used to diffeological spaces from a practical point of view? It seems to me that papers on this topic are mostly concerned with their very good ...
3
votes
3answers
298 views

undergraduate handle decomposition. Reference

As the title says, I'm searching for a nice textbook for introducing the theory of handle decomposition of manifolds to undergraduate students.
3
votes
1answer
210 views

Exponential mapping versus flow

In Hamilton's article on the Nash-Moser Theorem, he gives the map that maps a vector field $X$ to its flow $e^{tX}$ in $\mathrm{Diff}(M)$ as an example where the implicit function theorem in Frechet ...
3
votes
1answer
203 views

Is any smooth homeomorphism isotopic to a smooth embedding?

Let $f:D^m\to \mathbb{R}^m$ be a smooth map ($D^m$ is the unit ball). We call $f$ embedding if it is a homeomorphism on the image and the derivative $D_xf$ is nonsingular at each point $x\in D^m$ ...
-1
votes
1answer
141 views

$S^n$ admit a real polarization $D\subset TS^n$?

When the $n$-sphere, $S^n$,admit a real polarization $D\subset TS^n$
4
votes
1answer
357 views

Quantization of symplectic vector space and choice of lagrangian subspaces

My question is related to Geometric Quantization. I don't undrestand the philosophy of following assertion If $(V,\omega)$ be a symplectic vector space then the quantizations of $V$ ...
2
votes
1answer
491 views

Is it possible to define a structure of differentiable manifold on the Hilbert cube $[0,1]^\mathbb{N}$?

Is it possible to define a structure of differentiable (smooth) manifold on the Hilbert cube $[0,1]^\mathbb{N}$ ? Has it been done in the literature? In textbooks, only the Banach case is treated, ...
2
votes
3answers
237 views

Fatou sets and topological entropy

Let us consider a diffeomorphism of a compact real manifold (complex manifold defined over the reals), and let us say that the diffeomorphism is birational. Hence, it extends to a birational map from ...
1
vote
1answer
262 views

The space of generalized complex structures in sense of N.Hitchin is contractible?

Generalized complex structures were introduced by Nigel Hitchin in 2002. A generalized almost complex structure is an almost complex structure of the generalized tangent bundle which preserves the ...
0
votes
0answers
99 views

transverse intersection of Whitney stratifications

Let $M$ be a smooth manifold. If $X$ and $Y$ are two Whitney objects, i.e. subsets with a given Whitney stratification, then $X$ and $Y$ are transverse if each stratum of $X$ is transverse to each ...
3
votes
2answers
270 views

Is it impossible for the dimension of a topological space to increase under a smooth map?

First let me make a definition. Let $M$ be a smooth manifold and $S \subset M $ a topological subspace of $M$. We say that $S$ has "dimenion" at most $k$ if $S$ is a subset of $$ X_1 \cup X_2 \ldots ...
3
votes
1answer
109 views

Hilbert Manifolds and embedding

In the Wikipedia article on Hilber manifolds, it is claimed that every Hilbert manifold can be smoothly embedding onto an open subset of the model Hilbert space. However, no explicit reference is ...
10
votes
1answer
617 views

Does the gluing procedure in Robert Wald’s book *General Relativity* yield a Hausdorff spacetime?

Before I state my problem, let me provide some definitions pertaining to the Cauchy Problem in General Relativity. Definition 1: A triplet $ (\Sigma,h,k) $ is called an initial data set if $ ...
0
votes
1answer
116 views

A version of implicit function theorem when sections are not everywhere smooth?

Let $V_1, V_2 \rightarrow M $ be smooth vector bundles over a manifold $M$ and $s_1: M \rightarrow V_1$ a smooth section transverse to the zero set and $s_2: M \rightarrow V_2$ a continuous section ...
1
vote
0answers
136 views

Can one always extend a smooth section defined on a non compact submanifold to the whole manifold, provided it extends continuously to the closure?

Let $V \rightarrow M$ be a smooth vector bundle over a smooth compact manifold $M$ (without boundary) and $X \subset M$ a smooth submanifold of $M$, that is not necessarily closed. Suppose $s: X ...
2
votes
0answers
75 views

Squared displacement function on Lorentzian manifolds

Hi, Let $\varphi$ be an isometry of a simply connected pseudo Riemannian manifold $M$. The squared displacement function of $\varphi$ is $d^2_{\varphi}(p):=d^2(\varphi (p),p)$, $p\in M$, where $d$ is ...
2
votes
1answer
345 views

Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space?

Suppose that $X$ is a compact, finite dimensional manifold and $Y$ is an infinite dimensional, second countable ($C^\infty$-)Banach manifold. Let $\nu \in \mathbb{N}$. Question: Is the space ...
12
votes
1answer
569 views

Strong Whitney embedding theorem for non-compact manifolds

$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds. The strong ...
4
votes
1answer
482 views

Differentiable manifolds by Serge Lang question

I have started reading "Introduction to differentiable manifolds" by Serge Lang. In this book, Lang takes a different approach, by immediately introducing manifolds on arbitrary Banach spaces. His ...
2
votes
1answer
152 views

Vector fields on a simplicial manifold.

Is there a known definition of vector fields on a simplicial manifold? For me, it seems natural that the definition should be something along the lines: Let $M_{\bullet}$ be a simplicial manifold ...
1
vote
1answer
158 views

Can elements of Weil algebras be detected by maps into truncated symmetric algebras?

Recall that a Weil algebra is a finite-dimensional real unital algebra that admits exactly one homomorphism to R. Such algebras form the basis of the Weil approach to differential geometry, pioneered ...
2
votes
2answers
309 views

if $S \times \Re$ is diffeomorphic to $T \times \Re$ then are S and T diffeomorphic?

Suppose that $S$ and $T$ are two smooth manifolds and '$ \Re$' be the reals with the normal manifold structure. And here I use '$=$' to mean diffeomorphism. Is the statement below true? $ S \times ...
1
vote
1answer
140 views

finding effective 2-form corresponding to an equation

What is the effective 2-form corresponding to the equation $det Hess v=(v-q_1v_{q_1}-q_2v_{q_2})^4$ you can find the definition of effective forms here
10
votes
3answers
499 views

Characterizing Hessians among symmetric bilinear tensors

I apologize in advance if this is somewhat elementary, but: Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\in Sym^2(M)$ are ...
1
vote
2answers
312 views

Is this surface diffeomorphic to a sphere(S^2)? [closed]

let $f:R^3 -> R \ \ \ \ \ \ \ \ f(x,y,z)=x^4 + y^6 +z^8 \\$ $M = f^{-1}(1)$ Is M is diffeomorphic to a sphere $S^2$ ? I tried to solve this problem, but I realized that I have no tools to ...
0
votes
1answer
225 views

A $C^\infty$-function on a submanifold which is not the restriction of a a $C^\infty$ on $M$

I am looking for an example showingthat a function $f$ which is $C^\infty$ on a submanifold $N$ of $M$, but it cannot be written as the restriction of a $C^\infty$-function on $M$.
3
votes
2answers
235 views

Real analytic submanifolds of $\mathbb{R}^{n}$

Hallo, Let $(M,g)$ be a Riemannian $k$-dim real analytic submanifold of $\mathbb{R}^{n}$. Is it true that $M$ in $\mathbb{R}^{n}$ looks locally (in a small neigbourhood around some point in $M$) as ...
4
votes
1answer
349 views

What manifolds can have a (non-piecewise) linear structure?

By the definition I'm using, all manifolds are Hausdorff and second countable. For all non-negative integers $n$, I define $B_n$ to be $\bigl\{ \mathbf{v} \in \mathbf{R}^n : \lVert\mathbf{v}\rVert ...
3
votes
2answers
434 views

Rotation in Hyperkähler manifolds

Any Hyperkähler manifold has 3 complex structures $I_{1}, I_{2}, I_{3}$. Assume that there is an additional complex structure $J$. Can this be written as $J = aI_{1} + bI_{2} + cI_{3}$, where $(a,b,c) ...
4
votes
2answers
359 views

Do transvers foliations induce complex structure?

Hallo, I have the following question: Let $M$ smooth analytic manifold of dimension 4n. Assume furthermore that $M$ admits two foliations $A$, $B$, both with leaves of dimension 2n such that the ...
3
votes
1answer
193 views

Holonomy of a Kähler manifold

Hi, I have the following question: Let $(M,J, \omega)$ be a Kähler manifold (not necessary compact). We know that the holonomy group is a subgroup of $U_{n}$. Let $\Omega$ be a constant ($\nabla ...
1
vote
0answers
173 views

HyperKaehler manifolds are Ricci-flat

Hi, I have the following question: Let $M$ be a Hyperkaehler manifold with complex structures $I,J,K$ and Hyperkaehler metric $g$. Let $\omega_{I} = g(I *, *), \omega_{J} = g(J *, *), \omega_{K} = ...
1
vote
1answer
266 views

Holonomy group of a non-compact Kaehler manifold

Hallo, I have the following question: Let $(M,I,\omega)$ be a not necessary compact Kaehler manifold of complex dimension $n$. Assume that there exists a nowhere vanishing holomorphic $(n,0)$-form ...
0
votes
1answer
401 views

Geodesics on a twisted torus

This is a repost of a question I posted at MSE. Mark L. Irons' paper The Curvature and Geodesics of the Torus gives a concise overview of the geodesics on the torus: There are five clear-cut ...
2
votes
2answers
316 views

Isometric embedding of a Kaehler manifold as a special Lagrangian in a Calabi-Yau manifold

Hallo, I am reading the paper "Hyperkaehler structures on the total space of holomorphic cotangent bundles" by D.Kaledin and I am asking if it is possible to embedd a real-analytic Kähler manifold, ...
2
votes
1answer
116 views

Can stabilizer groups in an orbifold have global twisting?

Can stabilizer groups in an orbifold have global twisting? For example, consider the two groups $\mathbb Z/3\times\mathbb Z$ and $\mathbb Z/3\rtimes\mathbb Z$ (where $\mathbb ...
0
votes
1answer
212 views

Unique symplectic form in an adapted complex structure

Hallo, I ave the following question: Due to Stenzel, Lempert, Szöke ect. we know that a Riemannian manifold $(M,g)$ admits a complex structure on an neighbourhood of the cotangent bundle. This ...
4
votes
0answers
188 views

Ricci-flat metrics on Cotangent bundles in adapted complex structure

greetings, Let $(M,g)$ be a compact Riemannian manifold. On some neighbourhood $X$ of the zero section in the cotangent bundle $T^{*}L$ we have a complex structure $J$ and a Kähler form $\omega$ s.t. ...