For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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2
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31 views

$V$ is $C^1$ and $V(x_0)=0$ and $ \nabla V $ is not zero $\{ x : V(x)= c \}$ is a surface with no edge around $x_0$

I am studying lyapanov second method in stablity theory of ODE. I have encountered a geometric lemma which says the following: Assume $ V:\mathbb R^n \to \mathbb R$ is a $C^1$ and $x_0 \in \mathbb ...
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1answer
33 views

Poincaré invariant corresponds to area?

In this paper(equation 34) click me it says that if we have a symplectic egg, then the area of a $(x_i,p_i)$ clone intersected with the egg is given by the poincaré invariant $\int p dx = \int_0^{2 ...
3
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1answer
123 views

Perturb a given smooth function to a Morse function relative to fixed level sets, which are already fine.

Let $M$ be a manifold (not necessarily compact) , for the sake of clearness embedded in $\mathbb{R^n}$ and $f\colon M\rightarrow \mathbb{R}$ a smooth function. The theorem of Sard gives us that ...
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0answers
29 views

A manifold with boundary in $\mathbb{R}^{n}$.

I want to show that the cylinder $C = \{ (x,y,z)\in\mathbb{R}^3: x^2 + y^2 = 1, 0 \le z \le 1 \}$ is a differentiable manifold with boundary, of dimension 2, this is: A subset $M \subset ...
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1answer
52 views

Show that this is indeed a differentiable manifold with boundary.

I want to show that the cylinder: $$C = \{ (x,y,z)\in\mathbb{R}^3: x^2 + y^2 = 1, 0 \le z \le 1 \}$$ is indeed a a differentiable manifold with boundary, this means the following: A subset $M ...
0
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2answers
65 views

Local representation of a submanifold as a graph over the tangent plane

I'd like to verify the following statement, which intuitively seems quite reasonable, by a rigorous proof: Let $M \subset \mathbb{R}^D$ be a $d$-dimensional $C^1$ submanifold embedded in ...
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0answers
22 views

Isomorphism of the Clifford bundle of a Riemannain manifold

Let $(M,g)$ be an oriented Riemannian manifold and $Cl(M):=\bigcup_{x\in M}Cl(T_xM,g_x)$ be the clifford bundle of $(M,g)$. (Here $Cl(T_xM,g_x)$ denotes the clifford algebra of the vector space ...
2
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1answer
77 views

Sheaf, étalé space with Riemann surfaces.

Let $f:X\rightarrow Y$ be an holomorphic map betwen two Riemann surfaces and let: $\Gamma:=${ $(x,y)\in X\times Y|y=f(x)$ } $\subset X\times Y$ be the graph of $f$. I have to show that ...
4
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1answer
55 views

$\omega$ is $1$-form on $S^1$.

Let $h: \mathbb{R} \to S^1$ be $h(t) = (\cos t, \sin t)$. How do I show that if $\omega$ is any $1$-form on $S^1$, then$$\int_{S^1} \omega = \int_0^{2\pi} h^*\omega?$$
1
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1answer
95 views

For $(n-1)$-form $\omega$ on $M^{n}$ compact, orientable without boundary, then $d\omega$ vanish for some point

Let $M^{n}$ manifold compact, orientable without boundary and $\omega$ $(n-1)$-form then there is $p\in M$ such that $d\omega(p)=0$. This is for my homework of integration on manifolds & Stokes ...
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2answers
56 views

Explain why this set is not a differentiable manifold

I want to figure out why the set of zeros of the function $g:\mathbb{R}^{2} \to \mathbb{R}$ defined as $g(x,y) = x^2 - y^2$ is not a differentiable manifold. So what I want to use is the following ...
2
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1answer
95 views

Eigenvalues of a Plane Curve Laplace-Beltrami Operator

Given a closed plane curve $C$, which is a one-dimentional manifold, what are the eigenvalues of Laplace-Beltrami operator defined on this curve? I know that the LB eigenvalue problem for unit ...
0
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1answer
30 views

Show that the tangent space of a manifold is a certain set.

Let $A\subset \mathbb{R}^n$ an open set, and $g:A\to \mathbb{R}$ continously differentiable such that $g'(x)\not=0 $ for $x\in A$. If $M = g^{-1}(\{0\})\not=\emptyset$, then I want to show that the ...
2
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1answer
95 views

Higher homology groups of infinite cyclic cover

Prove that all homology groups of the infinite cyclic cover of a knot complement are trivial except $H_1$. I've posted an answer below using Mayer-Vietoris. If you know of other arguments, please ...
1
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2answers
84 views

Why the set $g^{-1}(\{0\}) $ is not a differentiable manifold?

Let $g:\mathbb{R}^2 \to \mathbb{R}$ given by $g(x,y) = x^2 - y^2$. Then I am triying to figure out why this function is not a differentiable manifold , I was trying to give an explicit coordinate ...
3
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4answers
179 views

Are there compact manifolds without boundary?

Based on this question I'd like to know: Are there compact (sub)manifolds without boundary in $\mathbb{R}^n$ ? Because, as that question shows, the topology of the manifolds has to be the trace ...
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0answers
22 views

Pipe-fitting conditions in 3D

Let's we have smooth (continuous and infinitely differential) curve $f(x(t), y(t), z(t)) = 0$ in 3D. Now I want to build a pipe of diameter $D$ around it. Questions: What are the set of conditions ...
6
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1answer
149 views

Tautological line bundle over $\mathbb{RP}^n$ isomorphic to normal bundle? Also “splitting” of transition functions

Hallo fellow mathematicians. I try to understand why the normal bundle of $\mathbb{PR}^n$ is isomorphic (in the category of vector bundles) to the tautological line Bundle. More aptly, why ...
0
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1answer
39 views

The definition of C^r Structural Stability

I currently have a definition that states that given a flow $f$, $f$ is structurally stable if for any $g$ in some neighborhood of $f$, $f$ and $g$ are topologically conjugate. Would the definition ...
4
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0answers
31 views

What does it mean for a function f to be C^k “close” to a function g?

My impression is that $f$ is $C^k$ "close" to $g$ if $f$ is close to $g$, $f'$ is close to $g'$, ... $f^{(k)}$ is close to $g^{(k)}$. However, my professor is being a bit nebulous on what "close" ...
2
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1answer
51 views

Explaining problem in Gadea's “Analysis and Algebra on Differentiable Manifolds”

I have a lot of trouble trying to explain to myself what the author did in problem 1.102 (the answer is in the link): Let $TM$ be the tangent bundle over a differentiable manifold $M$. Let ...
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1answer
48 views

Codifferential and corresponding homology theory

This is the kind of a natural question which can come to mind after completing the standard course in differential geometry and homology theory: lety us start with a smooth manifold $M$. One can ...
4
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1answer
57 views

Nonorientable manifolds being a boundaries

I will say that a manifold (smooth, compact, without boundary) is itself boundary if there is some smooth compact manifold $W$ with boundary, such that $M=\partial W$. I'm interested in nontrivial ...
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1answer
64 views

Meaning of conditions on definition of topological manifold

This is my first time to study on the manifold. I've studied that the topological manifold is: Hausdorff, locally homeomorphic to Euclidean space, and second countable topological space. With ...
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1answer
52 views

Grassmanians and boudaries of manifolds

Let $M$ be a smooth, compact manifold without boundary. I will say that $M$ is a boundary when there is a smooth, compact manifold with boundary $W$ such that $\partial W=M$. After some lectures I ...
5
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1answer
40 views

G-P Exercise, immersion except at origin, what does its image look like?

(This is not a duplicate of another question on math.stackexchange, as that other question just basically asks for the answer to the question below, of which I have provided an answer to. My question ...
2
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0answers
23 views

Can the same surface have minimal genus in both a 3-manifold and a 4-manifold?

By a surface of minimal genus I mean in it's homology class: A surface $S_0$ embedded in a smooth manifold $M$ such that any other surface $S$ with $[S]=[S_0]\in H_2(M)$, we have $g(S)\geq g(S_0)$. ...
3
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1answer
49 views

Find a surface that has positive constant curvature that is not open subset of sphere

Can some one find a surface that has positive constant curvature that is not open subset of sphere. I know every connected and compact surface with positive constant curvature is sphere. I need ...
5
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1answer
105 views

Reference request: infinite-dimensional manifolds

The following books and/or notes develop various aspects of the theory of infinite-dimensional manifolds: Lang, Fundamentals of Differential Geometry. Kriegl & Michor, The Convenient Setting of ...
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1answer
42 views

First exercise of Guillemin-Pollack. [closed]

If $k < l$ we can consider $\mathbb{R}^k$ to be the subset $\{(a_1, \dots, a_k, 0, \dots, 0)\}$ in $\mathbb{R}^l$. Show that smooth functions on $\mathbb{R}^k$, considered as a subset of ...
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0answers
25 views

Area between two curves as manifold with boundary

Let $U \subset \mathbb{R}^n$ be open set, $F,G:U \to \mathbb{R}$ smooth function such that $F(x)<G(x)$. We define: $$\Omega=\{(x,y) \in U \times \mathbb{R}:G(x) \leq y \leq F(x)\}$$ I would like ...
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1answer
53 views

Boundary of a topological manifold invariant?

Let $M=(X,\tau)$ be a topological manifold with boundary. One can proof that the interior $Int(M)$ and boundary $\partial M$ of the manifold are distinct sets. I was wondering if someone knows a ...
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0answers
30 views

global manifolds

Can you also explain why the global stable manifold is the union of the flow of the local stable manifolds for t < 0? Why do we not include all t?
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1answer
71 views

Let $M\subseteq \mathbb{R}^k$: Manifold topology vs. trace topology

I'm confused about the topology of submanifolds of $\mathbb{R}^n$: Let $M$ be such a $k$-manifold (say, the circle $S^1$, of dimension $1$, embedded in say $\mathbb{R}^7$); the topology of such a ...
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0answers
29 views

About a map between two topological manifolds with different dimensions

Let $M_1$ be a $n$-dimensional topological manifold and let $M_2$ be a $m$-dimensional topological manifold, such that $m>n$. Moreover, let $U\subset M_1$ be an open set and let $f:U\rightarrow ...
3
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0answers
47 views

What can I say of an $m$-dimensional submanifold $S$ of an $m$-dimensional manifold $M$? [duplicate]

I consider a differentiable manifold $M$ of dimension $m$. Let be $S$ a submanifold of $M$ of the same dimension $m$. What can I say about $S$? I have tried to prove that $S$ is open but I get ...
0
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0answers
124 views

how to prove torus as a 2 dimensional manifold

Consider equation of torus in 3 dimension $(R-\sqrt{(x^2+y^2)})^2+z^2 = r^2$ where $R$ is larger radius and $r$ is smaller radius. how to prove that it is 2 dimensional manifold? I tried ...
2
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1answer
40 views

Prove that $X=\{(x,y,z,t) | x^2+6y^2+4z^2+t^2=1 \}$ is compact and connected

Let be $X=\{(x,y,z,t) | x^2+6y^2+4z^2+t^2=1 \}$. I have proved that $X$ is a submanifold of $\mathbb{R}^4$ of dimension $3$. I have to prove that $X$ is compact and connected. My idea, thinking of ...
3
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1answer
57 views

Integration with 2-forms

Wikipedia says: Let $$ \omega=f_{z}\, \mathrm dx \wedge \mathrm dy + f_{x}\, \mathrm dy \wedge \mathrm dz + f_{y}\, \mathrm dz \wedge \mathrm dx $$ be a 2-form on a surface with parametrization ...
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1answer
53 views

Quotiented manifold homeomorphic to a complex projective space?

I define an action on $\mathbb{C}-0 × \mathbb{C^2}-(0,0)$ by $(x,y,z) \mapsto ((1/a)x,ay,az)$ when $a$ is a non zero complex number, I get a manifold by quotienting. Taking element from this ...
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1answer
89 views

What is a real structure on a manifold?

I have been looking at manifolds (twistor spaces) that have a "real structure". I am not quite sure what this means. I've looked on Wikipedia and they have an article that explains real structures on ...
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1answer
36 views

Prove that $\mathbb{R}^2 \times S^1 $ and $M=\left\lbrace (x,y,z,w) \in \mathbb{R}^4 \mid x^2+y^2- z^2-w^2 = 1 \right\rbrace$ are diffeomorphic

Let be $M= \left\lbrace (x,y,z,w) \in \mathbb{R}^4 \mid x^2+y^2- z^2-w^2 = 1 \right\rbrace$. I have proved that $M$ is a embedded submanifold of $\mathbb{R}^4 $ of dimension $3$. I have now to ...
9
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2answers
102 views

Properties of the category of smooth vector bundles over a smooth manifold

I am wondering if there are any sources that discuss the properties of the category of vector bundles over a smooth manifold. It seems that most differential geometry texts I've looked at avoid ...
3
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1answer
138 views

How could a group be a manifold?

For example a Lie group is defined as a certain differentiable manifold, but what does this mean geometrically, and what is gained by viewing something abstract and algebraic as a manifold? First, I ...
2
votes
1answer
97 views

Is $S^1 \times S^1$ really a torus?

Consider a function $f(x)$ that is $2\pi$ periodic. Consider another function $g(y)$ that is also $2\pi$ periodic. If I wanted to compute the integral of either of these functions I would do so ...
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1answer
81 views

Retraction to the Boundary on Compact Manifold

I was given the following question on an exam today, "Suppose that $M$ is a compact $n$- dimensional oriented manifold with corners. A retraction to the boundary is a continuously differentiable map ...
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2answers
43 views

Properties of the ring of smooth function germs, question on proof.

Let us denote by $C_n$ the ring of $C^{\infty}$ smooth function germs $f : (\mathbb R^n, 0) \to \mathbb R$ or the ring of analytic functions germs $f : (\mathbb C^n, 0) \to \mathbb C$. Denote by ...
0
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0answers
93 views

An example of 4D Hypersurface in 3D

Number of combinations of 4 dimensions choosing 3 at a time is 4. Someone please give a description of a most elementary 4 Dimensional Hyper surface which has its four 3D intersections with ...
1
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1answer
80 views

Is $C = \{(x,y) \mid x^3 + xy + y^3 = c \} \subset \mathbb{R}^2$ an embedded submanifold of $\mathbb{R}^2$?

The problem As a continuation of this question (where it was shown that $C$ was a closed $1$-dimensional submanifold for $c \neq 1/27$), I'm trying to find out whether or not $$C = \{(x,y) \mid x^3 + ...
3
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1answer
89 views

Mean value theorem on Riemannian manifold?

Is there some generalisation of the classical mean value theorem for real-valued functions on an interval $$|f(x)-f(y)| \leq |\nabla f(c)||x-y|$$ for some $c$ between $(x,y)$ to the case where $f:M ...