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

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0
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1answer
30 views

Wedge product: prove $\omega \wedge \eta= (-1)^{kl} \eta \wedge w$

prove $\omega \wedge \eta = (-1)^{kl} \eta \wedge w$ where $\omega \in \Lambda^{k}(V)$ and $\eta \in \Lambda^{l}(V)$. This is from page 79 on spivak calculus on manifolds. my progress: $\omega ...
0
votes
1answer
42 views

Existence of pullback tensor

If $M,N$ are smooth manifolds and $F: M \to N$ is a surjective smooth submersion. A tangent vector $v \in T_p M $ is called vertical if $d F_p (v) = 0$. Now suppose $\omega \in \Omega^k (M)$, I want ...
0
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0answers
34 views

Construct circle as a submanifold of $\mathbb R^2$.

I have to construct the circle as a submanifold of $\mathbb R^2$. Let $f(x,y)=x^2+y^2$ which is $\mathcal C^\infty $. The circle is given by $$\mathbb S^1=\{(x,y)\mid x^2+y^2=1\}=f^{-1}(1).$$ We have ...
2
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2answers
23 views

Manifold and submanifold, in $S$ a submanifold?

I have a theorem that says that: Theorem : Let $M,N$ smooths manifolds and $f:M\to N$ a smooth application. If $Rg(f)=k$ in a neighborhood of $S=f^{-1}(q), q\in N$ then $S$ is a submanifold of ...
0
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0answers
31 views

Derivative of dual basis vectors in terms of Christoffel symbols

How can I demonstrate from $$ \frac{\partial \mathbf{e_j}}{\partial x^i} \equiv \Gamma_{ij}^k \mathbf{e_k} $$ what the value of $$ \frac{\partial \mathbf{e^j}}{\partial x^i} $$ (with the index now ...
3
votes
1answer
55 views

Is $C[0,1]$ a manifold?

I know that $C[0,1]$, as a topological space induced by the metric $d(f,g)=\sup_x |f(x)-g(x)|$, is Hausdorff, second countable, and has cardinality same as $\mathbb R$. But is it a manifold? By ...
6
votes
3answers
582 views

Are all Lie groups Matrix Lie groups?

I have beard a bit about so-called matrix Lie groups. From what I understand (and I don't understand it well) a matrix Lie group is a closed subgroup of $GL_n(\mathbb{C})$. There is also the notion ...
1
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1answer
7 views

Different between several proposition - Manifold, plate function.

I have 3 different proposition and I really have problem to see in what they are that different. Let $M$ a smooth manifold. Prop 1 : Let $U\subset M$ an open and $C\subset U$ a compact. Then, ...
0
votes
2answers
33 views

The exterior derivate

I am trying to show that $d(a \wedge da)=0$ if $k$, the degree of k-form $a$ is even. I have said: $=da \wedge da + (-1)^k a \wedge d^2a$ I believe the first term is zero due to repeated indices and ...
0
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0answers
14 views

Classification of manifold of dimension 1

By my course, all manifold of dimension 1 is isomorphic to $(0,1), (0,1],[0,1)$ or $\mathbb S^1=\{x^2+y^2=1\mid x,y\in\mathbb R\}$. I was thinking of a curve in the plan, with a knot. (See picture) ...
2
votes
2answers
116 views

Why $\mathbb S^1\times \mathbb S^1$ is isomorphic to the torus?

Could someone explain why $\mathbb S^1\times \mathbb S^1$ is isomorphic to the torus ? I recall that $\mathbb S^1=\{ x^2+y^2=1\mid x,y\in\mathbb R \}$. To me it would be more something like the ...
0
votes
1answer
32 views

Smooth extension of a tangent vector

Let $(M,g)$ be an $n$-dimensional smooth Riemannian manifold and $p_0\in M$, $v_0\in T_{p_0}M$ with $|v_0|=1$. $\nabla$ is the Levi Civita connection. How can we construct a smooth vector field $X$ ...
3
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0answers
48 views

What's “ancient time”?

I've found a reference to "ancient time" from Google. It's mentioned in e.g. the book Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincare Conjecture, by Qi S. Zhang. E.g. The ...
2
votes
1answer
28 views

Is there restriction on the cardinailty of manifold?

I only know that the each point has some neighborhood with cardinality same as $\mathbb R$, but I have no idea about the cardinality of the whole manifold, must it have cardinality as $\mathbb R$? ...
1
vote
1answer
19 views

If $M$ a manifold and $p\in M$, why is there a neighborhood V s.t. $V\subset \overline V\subset W$ with $\overline V$ compact?

If $M$ a manifold and $p\in M$. Let $W\subset M$ a neighborhood of $p$. Why is there a neighborhood $V$ of $p$ s.t. $V\subset \overline V\subset W$ with $\overline V$ compact ? I would say (but with ...
2
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0answers
45 views

Are there more convenient charts than Riemannian normal coordinate chart?

Let $(M,g)$ be an arbitrary smooth Riemannian manifold of $n$ dimensional and $p_0\in M$. $\nabla$ denotes the Levi Civita connection. It is well known that there is a coordinate chart around $p_0$ ...
1
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0answers
37 views

Let $f\colon \Bbb R^n \to \Bbb R^n$ be the translation, $f(x)=x+a$. Show that $\deg(f)=1$

Let $U,V$ be connected open subsets of $\Bbb R^n$ and let $f\colon U \to V$ be a $C^{\infty}$ proper map. For all $w \in \Omega_c^n(V)$, $\int_{U}f^*(w)=\gamma \int_Vw$. Now define $\deg(f)=\gamma$, ...
0
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2answers
45 views

Show that there is a smooth function $\tilde f\colon M \to \Bbb R$, with $\tilde f_M=f$ and $\operatorname{supp}(\tilde f) \subset U$.

Let $M \subset U \subset \Bbb R^n$, where $M$ is closed set and $U$ is an open set. Show that if $f\colon M \to \Bbb R$ is a smooth function. Then there is a smooth function $\tilde f\colon M \to \Bbb ...
0
votes
0answers
10 views

Circle homeomorphic to $\hat {]0,1[}$ the compactification of Alexandrov.

I denote $\mathbb S^1=\{x^2+y^2=1\mid x,y\in\mathbb R\}$. So, if I understood well, $$]0,1[\cong \mathbb S^1\backslash \{(0,1)\}$$ (or to $\mathbb S^1\backslash \{(a,b)\}$ where $(a,b)\in\mathbb ...
0
votes
1answer
13 views

Show $\varphi(x_1,…,x_{n+1})=\frac{1}{1-x_{n+1}}(x_1,…,x_n)$ is surjective.

We consider in fact the stereographic projection. Let $$\mathbb S^{n}=\{x_1^2+...+x_{n+1}^2=1\mid (x_1,...,x_{n+1})\in\mathbb R^{n+1}\}$$ and $E=\text{span}(e_{n+1})^\perp$ where ...
2
votes
1answer
47 views

Components in the complement of compact subset in manifold

Let $M$ be a (smooth) connected manifold and $K \subset M$ a compact subset. Then $M \setminus K$ consists of a number of components. Let $(U_j)_{j \in \mathcal J}$ be the collection of bounded ...
0
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0answers
24 views

Show that $\partial M\cong \mathbb S^1$ where $M$ is the moebius strip.

Let $M$ the Moebius strip and $\mathbb S^1=\{x^2+y^2=1\mid x,y\in\mathbb R\}$ the circle. Show that $$\partial M\cong \mathbb S^1.$$ Let $\sim$ the equivalence relation $$(x,y)\sim ...
5
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0answers
52 views

$M$ is homotopy equivalent to $S^n$. [duplicate]

Let $M$ be a compact connected $n$-manifold without boundary, where $n \ge 2$. Suppose that $M$ is homotopy equivalent to $\Sigma Y$ for some connected based space $Y$. How do I see that $M$ is ...
0
votes
1answer
23 views

Find a $C^∞$ map which is a one-one onto, inverse is continous but not a diffeomorphism.

$X \subseteq \Bbb R^m$ and $Y \subseteq \Bbb R^n$ Definition The map $f:X \to Y$ is $C^∞$ if for every $p∈X$ ,$\exists$ a nhbd $U_p$ of $p \in \Bbb R_n$ and a $C^∞$ map $g_p:U_p→R^m$ such that ...
0
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0answers
50 views

Is any of these two groups a smooth manifold?

This question came to my mind while I was going through Iian B. Smythe's talk titled A Crash Course in Topological Groups. In the talk it is mentioned that, Lie group G is a group, which is also ...
6
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0answers
76 views

Higher math and statistics/probability

So I've heard that certain areas of statistics and probability use manifolds and results from analysis and topology. Given that I lack the background to see where manifolds would become useful in ...
3
votes
1answer
71 views

Top Cohomology group of a “punctured” manifold is zero?

The following question is from my algebraic topology exam which I was unable to solve. Let $X$ be a orientable connected closed $n$-manifold. Let $ p \in X$. Show that $H^n (X-{p},R)=0$, where $R$ ...
0
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1answer
24 views

Schubert cell decomposition and full flags

I am looking for a self-contained basic theory of Schubert-cells through finding the decomposition of the full flag $Fl_3(\mathbb C^3)$.
0
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0answers
18 views

Contraction of a vector form

I'm trying to make sense of this definition, but I cannot see why the resulting map is in a space of dimension $k-1$, surely as it is comprised of k vectors this maps a k-form to a (k+1)-form? I ...
7
votes
1answer
118 views

Does it follow that $Y$ is homotopy equivalent to $S^{n-1}$?

Let $M$ be a compact connected $n$-manifold (without boundary), where $n \ge 2$. Suppose that $M$ is homotopy equivalent to $\Sigma Y$ for some connected based space $Y$. Does it follow that $Y$ is ...
7
votes
1answer
124 views

If $M$ is a nonorientable $3$-manifold, why is $H_1(M, \mathbb{Z})$ infinite? [duplicate]

Let $M$ be a compact connected $3$-manifold with boundary $\partial M$. If $M$ is nonorientable and $\partial M$ is empty, then how do I see that $H_1(M, \mathbb{Z})$ is infinite?
3
votes
1answer
45 views

Multiplication by $q$ annihilates $H_i(M, \mathbb{Z})$ if $1 \le i \le n - 1$?

See here for related. Let $M$ be a compact connected $n$-manifold (without boundary), where $n \ge 2$. Suppose $M$ is oriented with fundamental class $z$. Let $f: S^n \to M$ be a map such that ...
2
votes
1answer
30 views

$f_*: H_*(S^n, \mathbb{Z}_p) \to H_*(M, \mathbb{Z}_p)$ an isomorphism if $p$ is a prime that does not divide $q$.

Let $M$ be a compact connected $n$-manifold (without boundary), where $n \ge 2$. Suppose $M$ is oriented with fundamental class $z$. Let $f: S^n \to M$ be a map such that $f_*(i_n) = qz$ where $i_n ...
0
votes
1answer
23 views

is this map invertible?

Let $f: \Omega \to \mathbb{R}$ with $\Omega \subset \mathbb{R^n}$ open be smooth. Define: $g: \mathbb{R^n} \to \mathbb{R^{n+1}}$ by $g(x) = (x,f(x))$ does the map $g^{-1}|_{M\cap U}$ exist and is it ...
0
votes
0answers
27 views

Showing the determinant is non-zero

Let $f: \Omega \to \mathbb{R}$ be smooth with $\Omega$ open. Let $M = \{ (x,y) \in \mathbb{R}^n \times \mathbb{R} : y = f(x) \}$. Show that the following holds: there exists an open set $U$ with $x ...
2
votes
0answers
47 views

Lee's Topological Manifolds simplicial vs cw [closed]

Is it worth the time to read the first edition of Lee's topological manifolds to get his take on the simplicial complexes or not worry about it and only read the second edition which focuses on CW ...
1
vote
1answer
36 views

Schubert Cells of Flags

I have been reading on these notes Undergraduate Lectures on Flag Varieties and I need some explanations on two things: In page 3, how he modefied the matrices in the "Second Attempt" In the same ...
3
votes
2answers
76 views

$3$-manifold has same homology groups as a $3$-sphere. [closed]

Let $M$ be a compact connected $3$-manifold with boundary $\partial M$. If $M$ is orientable, $\partial M$ is empty, and $H_1(M; \mathbb{Z}) = 0$, does it follow that $M$ has the same homology groups ...
3
votes
1answer
34 views

Dimension of homology of boundary is twice the dimension of kernel induced by inclusion.

Let $M$ be a compact connected $n$-manifold with boundary $\partial M$, where $n \ge 2$. Assume that $M$ is orientable. Let $n = 2m + 1$ and let $K$ be the kernel of the homomorphism $H_m(\partial M) ...
4
votes
1answer
212 views

Compact $n$-manifold has same integral cohomology as $S^n$?

Let $M$ be a compact connected $n$-manifold without boundary, where $n \ge 2$. Suppose that $M$ is homotopy equivalent to $\Sigma Y$ for some connected based space $Y$. Does $M$ have the same integral ...
3
votes
1answer
83 views

Tangent spaces, how are vectors parallel transported?

I understand that tangent vectors lie in separate tangent spaces based on the point on which they are tangent to a manifold, but what about vectors that are parallel transported? For any manifold ...
0
votes
1answer
12 views

Path on a manifold has interval contained inside open ball

Let $M$ be an $m$-dimensional manifold, and let $g:[a,b]\rightarrow M$ be a path. Let $t\in(a,b)$. There exists a neighborhood of $g(t)$ in $M$ that is homeomorphic to an open ball $B(x,\epsilon)$ in ...
1
vote
1answer
20 views

Check that F*(Xp) is a derivation at F(p)

To show linearity is simple but I am stuck on derivation
1
vote
1answer
42 views

Product of smooth maps from $M \to \mathbb{R}$ is smooth

In Lee's Introduction to Smooth Manifolds there is a exercise to proof that if $f: M \to \mathbb{R}$ and $g: M \to \mathbb{R}$ are smooth, so is $fg$. The question I'm having is the following: By ...
1
vote
1answer
43 views

embeddings of projective spaces into Euclidean spaces

Let $\mathbb{R}P^n$, $\mathbb{C}P^n$, $\mathbb{H}P^n$ be the real, complex, quaternionic projective spaces resp. I want to find all $n$ such that $\mathbb{R}P^n$ can be embedded into ...
0
votes
2answers
23 views

Existence a path of a smooth manifold

Given a continuous differentiable functio $F:\mathbb{R}^n\mapsto \mathbb{R}^m$ with $n>m$. Define $$ {\cal M}=\{x\in\mathbb{R}^n: F(x)=0\} $$ and let $x_0\in{\cal M}$ such that the Jacobian of ...
0
votes
1answer
25 views

Example of germs not involving series

note: moved from mathoverflow, as off topic. I'm currently reading this book: http://www.springer.com/us/book/9781441973993 And when speaking about germs of functions the only example provided of two ...
6
votes
2answers
95 views

(Anti-) Holomorphic significance?

What are holomorphic and anti-holomorphic components? Why don't we call them complex components and their conjugates? What is holomorphic coordinate transformation?
7
votes
1answer
44 views

Is the union of an increasing sequence of topological copies of $\mathbb{R}^n$ homeomorphic to $\mathbb{R}^n$?

Let $M$ be an $n$-dimensional topological manifold, and let $(U_k)_{k \in \mathbb{N}}$ be an increasing sequence of open sets $U_k \subset M$ such that for each $k \in \mathbb{N}$, $U_k$ is ...
2
votes
1answer
28 views

Is the set of points where two functions agree a submanifold?

Let $M, N$ be smooth manifolds, $f : M \to N$ a smooth submersion and $g : M \to N$ a smooth function. Is it true that $R = \{x \in M\ |\ f(x) = g(x)\}$ is an embedded submanifold of $M$? How about if ...