For questions about smooth manifolds.

learn more… | top users | synonyms

1
vote
0answers
10 views

Oriented Hypersurface admits normal vector field

This question is the converse of this question and is taken from Lee's Smooth Manifolds, problem 15.7. Namely: Suppose $M$ is an oriented Riemannian manifold and $S\subset M$ is an oriented smooth ...
-1
votes
0answers
19 views

How to show that $RP^n$ cannot be obtained as the pre-image of a regular value of any smooth map $\phi : RP^{n+1} \rightarrow R$?

How to show that $RP^n$ cannot be obtained as the pre-image of a regular value of any smooth map $\phi : RP^{n+1} \rightarrow R$?
1
vote
1answer
21 views

showing pushfarward

Let $M,N$ be two differentiable manifolds and $f:M \rightarrow N$ be a smooth map. Define a new map $F:M\rightarrow M\times N$ by $F(p)=(p,f(p))$ I can prove first part which is F is smooth but I can ...
1
vote
0answers
13 views

An expression of covectors acting on vectors on the tangent space of a manifold

Let $M$ be a smooth manifold. Take $p\in M$ and $(U,\varphi)$, $\varphi:U\rightarrow \mathbb{R^n}$, a chart around $p$. Let $\mathbb{R}^n\left[\frac{\partial}{\partial x_i}\right]$ and ...
1
vote
1answer
27 views

How to show that two vector fields commute?

Could anyone help me with how to start to solve the following problem? From this problem as well as this, I have: Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such ...
0
votes
0answers
27 views

Vector fields that are smooth on the open unit ball $B$, are smooth on $\mathbb{R}^n$ if they are zero outside $B$

Could anyone help a little with the following problem? It is a continuation of this problem, but I will restate the things that are needed: Fix $\varepsilon \in (0, 1)$ and choose a smooth ...
3
votes
0answers
96 views
+100

Complex submanifold has the minimal volume

I know that the following theorem is true: If $W$ is a purely $k$-dimensional analytic subvariety of a domain in $\mathbb{C}^n$, $sngW$ is the set of its singular points, $V \subset W$ is open, ...
3
votes
1answer
40 views

How to show that $f : \mathbb{R}^n → \mathbb{R}^n$, $f(x) = \frac{h(\Vert x \Vert)}{\Vert x \Vert} x$, is a diffeomorphism onto the open unit ball?

Could anyone help me with the following problem? The problem Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for ...
0
votes
0answers
19 views

Showing that the 2-torus is parallelizable

Here is the question Let $$ \widehat{\xi}: \mathbb{R}^2 \to \mathbb{R}^2 $$ be a smooth function satisfying $$ \widehat{\xi}(x,y)=\widehat{\xi}(x+m, y+n) $$ for all $x,y\in \mathbb{R}, ...
1
vote
2answers
43 views

What does it mean to say a diagram commutes?

$\require{AMScd}$ In the context of smooth manifolds, the map $F:M\rightarrow N$ is smooth if $G$ on the below diagram is smooth. $\begin{CD} M @> F > > N\\ @V \varphi V V @V V\psi V\\ ...
1
vote
1answer
27 views

Are PL-homeomorphic manifolds diffeomorphic?

Take two smooth manifolds. Since they are smooth, they both possess triangulations. Now assume that the triangulations are related by Pachner moves, that is, the triangulated manifolds are ...
2
votes
0answers
37 views

How to prove that the tangent map $T\phi$ into the pullback bundle is smooth?

Assume $\phi: M\rightarrow N$ is smooth. Let $\phi^*(TN)$ be the pullback bundle of $TN$ by $\phi$. Define $T\phi:TM\rightarrow \phi^*(TN)$ as follows: $T\phi(m,v)=(m,d\phi_m(v)) $. We also have the ...
1
vote
1answer
20 views

Fractional Sobolev spaces on closed manifolds

Let $M$ be a closed manifold and $0<s<1$. How is the fractional Sobolev space , $H^s(M)$ defined? In particular if $M$ is a closed smooth simple curve in the place how is $H^{1/2}(M)$ ...
2
votes
0answers
40 views

Metric tensors and linear (differential) operators defined on a manifold and its osculating sphere

Given a 1D Riemannian manifold $\Gamma$ embedded in 2D Euclidean space (e.g. a parametric curve on a plane $\mathbb{R}^{2}$ ), and point $x_{0}\in \Gamma$, we denote $S^{1}(x_{0})$ the circle ...
3
votes
1answer
67 views

(co)limits in the category of diffeological spaces vs. category of smooth manifolds

I am wondering which (co)limits that exist in the category of smooth manifolds are preserved by the inclusion into the category of diffeological spaces? Are there any results that allow us to ...
7
votes
1answer
78 views

A “Theorem Style” Problem Book in Differential Geometry

I am trying to teach myself differential geometry using Lee's Introduction to Smooth Manifolds. To test my understanding, and learn the subject better, I am looking for a good problem book in ...
1
vote
0answers
32 views

Product of Two Orientable Manifolds is Orientable

I am trying to show that following: Let $M$ be an oriented smooth manifold of dimension $m$, and $N$ be an oriented smooth manifold of dimension $n$. Then $M\times N$ is orientable. Let ...
1
vote
1answer
34 views

Clarification on basic (horizontal) differential forms

Here's a question from Lee's Smooth Manifolds (Exercise 12-9) which was more or less answered here. The question is this: Let $\pi:M\to N$ be a smooth surjective submersion between smooth ...
1
vote
1answer
29 views

Natural Isomorphism Between Two Desciptions of Tangent Space

Let $M$ be a smooth manifold. Let $(T_pM)_{alg}$ denotes the "algebraist's" tangent space at $p\in M$, that is the tangent space via derivations, and $(T_pM)_{kin}$ denote the "kinematic" or ...
0
votes
0answers
8 views

cocompact action + finite stabilizers => proper action?

Assume a disctere countable group G acts on a smooth manifold M by diffeomorfisms and (1) M/G is compact and Hausdorff (2) all stabilizers are finite How to prove that the action is proper?
0
votes
1answer
63 views

Coordinate expression for the divergent

Let $(M, g)$ be a Riemannian manifold. As in Lee's Riemannian Manifolds book, we define the divergent of a vector field $X \in \mathfrak{X}(M)$ by the identity $d(\iota_X dV) = (div X) dV$, where ...
0
votes
0answers
36 views

Flows on a compact smooth manifold.

Statement: Let $f$ be a real-valued smooth function on a compact n-manifold $M$.Suppose it have finitely many critical points $\left\{p_1,...,p_k\right\}$ with associated critical values ...
1
vote
1answer
44 views

Guillemin & Pollack's proof on Whitney embedding theorem

I am confused with a little detail in Guillemin & Pollack's proof on Whitney embedding theorem. Please see page 54 in their book "Differential topology". In the second paragraph of page 54, they ...
2
votes
0answers
18 views

Extending a vertical vector to a vertical vector field

Let's say $F: M \rightarrow N$ is a smooth submersion between manifolds. Then each fiber of $F$ is a properly embedded submanifold. If $S$ is such a fiber, and I take some derivation $D \in T_p S$, ...
0
votes
1answer
29 views

Finding coordinate charts from a torus (square with sides identified) to $\mathbb R^2$.

I have a torus which is thought of as the unit square with sides identified in the usual manner. I know that for every point on the square I have to find a neighborhood and a diffeomorphism to open ...
6
votes
1answer
70 views

Motivation for the definition of an infinitesimal object

An infinitesimal object $D$ in a Cartesian closed category $\mathsf{C}$ is one for which the internal Hom functor $$(-)^D: \mathsf{C} \to \mathsf{C}$$ has a right adjoint. I am wondering what is the ...
2
votes
0answers
30 views

Extending a function on a submanifold to the ambient manifold & proof of a property of a vector field.

$\newcommand{\wt}[1]{\widetilde{#1}}$ Hello, I just tried my hand at two exercises from John M Lee's book Riemannian Geometry and I would like to know whether my reasoning is sound or if I did ...
4
votes
0answers
61 views

Are there enough knots to cover $\mathbb{R}^3$? [closed]

Actually, several years ago I was in a short, introductory, course about knot theory, and my original question that I posed was: "can the knots be used to classify homeomorphims in $\mathbb{R}^3$?". ...
3
votes
1answer
55 views

Finding critical values of a function on an embedded surface

Prior to the problem, we have already shown that $\Sigma=\{x_1x_2^2+x_2x_3^2+x_3x_1^2=1\}\subset\mathbb{R}^3$ is an embedded hypersurface and that the function ...
3
votes
1answer
140 views

Is my attempt to define the concept “smooth manifold” as a structure satisfying certain axioms correct?

In the lecture notes for a class I'm currently taking, smooth manifold structures are defined as equivalence classes of atlases. However, the issue I'm having is that its not entirely clear (to me) ...
8
votes
1answer
114 views

Explain densities to me please!

When it comes to integration on manifolds, I speak two languages. The first is of course the language of differential forms, which is something I am relatively well acquinted with. The second ...
0
votes
1answer
27 views

Coordinate free definition of the canonical one-form

There is apparently a naturally defined one-form on the cotangent bundle of a smooth manifold $M$: We have the cotangent bundle $\pi:T^*M \rightarrow M$; taking its derivative gives $d \pi:TT^*M ...
4
votes
3answers
166 views

Are there such things as 'locally homogenous spaces'?

A Euclidean space has the property that every point has a neighbourhood that is homeomorphic to some neighbourhood of any other point. I'm not sure what the name of this property is - I thought it ...
1
vote
1answer
29 views

Help with old exam question relating a spanning set of covectors of $T_{p}^{*} M$ to a coordinate chart containing $p$.

This question came up on my exam yesterday, and I still can't seem to come up with the proof for it: Let $M$ be an $n$-dimensional smooth manifold and suppose that $y^1, \ldots, y^n \in ...
2
votes
2answers
119 views

Integrating over the two form

Let $A=(0,1)^2$. Let $\alpha:A\to\Bbb R^3$ be given by the equation $$\alpha(u,v)=(u,v,u^2+v^2+1)$$ Let $Y$ be the image set of $\alpha$. Evaluate the integral over $Y_\alpha$ of the 2-form ...
1
vote
1answer
137 views

Hermitian form, fundamental $2$-form of Kahler structure on $\mathbb{C}^n$

I've come across the following (it is an excerpt of Stolzenberg's lecture notes 19): Wirtinger's Inequality. Let $L$ be a complex linear space and let $M$ be a real even-dimensional ...
1
vote
0answers
27 views

How to prove that $\forall$ Riemannian manifold $M$, $T\in C^{1}(M)$ and $x,y\in M$ close enough: $d(Ty,exp_{Tx}(D_xTexp^{-1}_xy))\leq d(x,y)$?

How to prove that $\forall$ Riemannian manifold $M$, $T\in C^{1}(M)$ and $x,y\in M$ close enough: $d(Ty,exp_{Tx}(D_xTexp^{-1}_xy))\leq d(x,y)$ ? My attempts so far were only able to show the ...
1
vote
1answer
28 views

equivalent characterizations of differentiability of maps between smooth manifolds (a la Warner)

I have been trying to supply the details for why two different definitions of differentiability for continuous maps $\psi:M\to N $ between smooth manifolds are equivalent. Here are some background ...
0
votes
1answer
18 views

How can I show that the set of images of a parameterized curve is smooth?

I know that in order for a set S to be smooth, it needs to be connected and that for each point $\vec{a} \in S$, there needs to be a neighborhood $N$ such that $S \cap N$ is a class $C^1$ function. ...
0
votes
1answer
76 views

Good text on Differential manifolds?

I am new in field of topology.I am finding to read good self readable text on differential manifolds.
2
votes
1answer
160 views

Integration on manifolds with singular points, corners

I'm looking for interesting examples of application of Stokes theorem for manifolds with singularities/corners. The theorem was mentioned here: ...
0
votes
0answers
29 views

Finding boundary coordinate chart

I need to calculate the boundary coordinate chart of the manifold with boundary $$M=\{(x,y,z)\colon x^2+y^2+z^2=1, z\ge0\}$$ If I define $U=\{ (u,v)\colon u^2+v^2\lt1 \}$ and ...
1
vote
2answers
59 views

Proving that $\Delta(M \times M)$ is a submanifold of $M \times M$

I am struggling to prove that $\Delta(M \times M) = \{(x,x) : x \in M\}$ is a submanifold of $M \times M$. A manifold M is a submanifold of N if there is an inclusion map $i:M \rightarrow N$ ...
5
votes
1answer
60 views

Proving that given any two points in a connected manifold, there exists a diffeomorphism taking one to the other

Suppose M be a connected manifold and $x, y$ ∈ M are two points. Then I'm trying to show that there is a diffeomeorphism f of M that takes $x$ to $y$. Since the set of points for which there is a ...
1
vote
0answers
46 views

Germ induced by a submanifold

I'm currently reading this article: STOKES' THEOREM ON REAL ANALYTIC VARIETIES by LUTZ BUNGART In it the author writes: "In the following, we let $W$ be a closed real analytic subvariety of an open ...
0
votes
0answers
30 views

Is the determinant bundle the pullback of the $\mathcal O(1)$ on $\mathbb P^n$ under the Plucker embedding?

Let $V$ be a $n$-dimensional complex vector space and consider the Grassmannian of complex $k$-planes $Gr(k,V)$. The Plucker embedding is an embedding $p:Gr(k,V) \to \mathbb P^M$ where $M = ...
1
vote
0answers
36 views

Why derivations obey chain rule.

Let $X, Y, Z$ be smooth manifolds and suppose we have smooth maps $$ F:X\to Y, $$ $$ G:Y\to Z. $$ By derivation at $x\in X$ I mean a linear map $$ \mathfrak{d}:C^\infty(X) \to \mathbb{R}, $$ such that ...
3
votes
0answers
34 views

Can We Write the Differential in Terms of Covectors?

Let $f:\mathbf R^n\to \mathbf R$ be a smooth map. We can write $df:T\mathbf R^n\to \mathbf R$ neatly as $$ df = \sum_{i=1}^n(\partial f/\partial x_i) dx_i $$ For a function $f:M\to \mathbf R$ defined ...
2
votes
0answers
68 views

Smoothness and algebraicity

My question is the following : given a complex algebraic affine variety $M \subset \mathbb{C}^n$, let us assume that it has moreover a smooth differential structure as a submaifold of ...
2
votes
2answers
67 views

$TS^1$ is Diffeomorphic to $S^1\times \mathbf R$.

I know this is a very basic question. But I am unable to get every detail right. I need to show that $TS^1$ is diffeomorphic to $S^1\times \mathbf R$. (I am using the concept of derivations to ...