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

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In the def. of a (k+l)-manifold with boundary, is it permissible for the domain of the coord. patch to be open in R^k x H^l?

Studying Munkres's "Analysis on Manifolds" on my own. One exercise (#5, p. 103), asks the reader to show that if $M$ is a k-manifold without boundary in $R^m$, and if $N$ is an l-manifold in $R^n$, ...
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1answer
115 views

Is this a geodesic?

Let $(M,g)$ be a riemannian manifold. Let $p$ in $M$ and $v,v_{0}$ two vectors in $\mathrm{T}_{p}M$. I am looking at the curve $$ \gamma \, : \, t \, \longmapsto \, \mathrm{Exp}_{p}(tv+v_{0}) $$ ...
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1answer
54 views

Transferring a result in PDE from open domain to a manifold

Can anyone recommend me something that in detail, talks about transferring a result in Sobolev space (as opposed to Holder spaces or something like that) that holds for open domains in $\mathbb{R}^n$ ...
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2answers
162 views

References about 3-manifolds

I am working on a subject of geometric group theory closely related to 3-manifolds, and in order to understand these links, I am seeking a good reference about 3-manifolds, as self-contained as ...
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1answer
50 views

Isotropy group for the subset of Grassmannian

Consider a complex Grassmannian $Gr_{k}(C^{n})$, which is a symmetric space with symmetry group $U(n)$ (i.e. unitary group). Consider a subspace $S_{0}$ of the Grassmannian determined by the canonical ...
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1answer
64 views

Compact Implies Closed

In general, i believe, it is not true that Compact Implies Closed. At least it is true that Compact does not imply closed and bounded. However, in case of differentiable manifolds, since they are ...
3
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1answer
132 views

Homework: calculation about differential form

Here is the question: Let $\omega = A dy\wedge dz + B dz \wedge dx + C dx \wedge dy$ in $\mathbf{R}^3$, and $d\omega = 0$. Denote \begin{eqnarray} \alpha = \int_0^1 tA(tx,ty,tz)dt\cdot(ydz-zdy)\\ ...
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125 views

Supremum Definition in context of a Rectifiable/Differentiable Curve

I am trying to prove that if I have 2 paramterizations of the same curve $\gamma$ and $\sigma$ (i.e. there is continuous bijective map $\phi$ such that $\sigma = \gamma \circ \phi$) then if the curve ...
2
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2answers
107 views

Using a non-zero wedge product to write a set of vectors as a linear combination of another set of vectors in a finite dimensional space.

Question: Let $V$ be a finite dimensional vector space, and let $ \{ v_1, ..., v_r\}$ and $\{w_1, ..., w_r\}$ be two sets of vectors in $V$. Suppose that $\sum_{i=1}^{r} v_i \wedge w_i = 0$, and ...
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2answers
92 views

Proving a submanifold of $SL_2(\mathbb{R})$

I already showed that $SL_2(\mathbb{R})$ is a 3-dimensional manifold. Now I want to show that the subspace $E$ of symmetric matrices whose eigenvalues are positive in $ SL_2( \mathbb{R})$ is a ...
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2answers
315 views

Grassmannian, Plucker coordinates

In which books can I find something about the grassmannian and the plucker coordinates ?
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1answer
100 views

How to obtain a diffeomorphism between $\mathrm{SL}(2,\mathbb{R})$ and $ (\mathbb{C}\!\smallsetminus\!\{0\})\times\mathbb R$?

Could someone please give me a tip on how to show that the map $\mathrm{SL}(2,\mathbb{R}) \to (\mathbb{C}\setminus\{0\})\times\mathbb{R}$ \begin{equation}\begin{pmatrix} a & b\\ c & d ...
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1answer
60 views

A Manifold Contained in Another

QUESTION: Let $M$ be a $k$-manifold-without-boundary in $\mathbb R^n$ and $N$ be another manifold-without-boundary in $\mathbb R^n$ such that $M\subseteq N$. Assume that there exists a point $\mathbf ...
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64 views

Proving $d(f^*e_i')=0$

Let $f:M\to N$ be a differentiable map between two manifolds. $e_i$ is a basis vector of $N$ with respect to some chart and $e_i'$ its dual (i.e. $e_i'(e_j)=\delta_{ij}$). How do I prove the ...
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1answer
103 views

Morse height function for general compact manifold

Can you give me the form of the height function for any compact manifold embedded in the reals? Maybe the projection of the parametrization onto a basis vector ex. For the n-sphere is ...
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2answers
103 views

complement of a finite subset of a path-connected space is path-connected

Given a smooth connected manifold $X$ of dim $\geq 2$, I need to show that $X\setminus Y$ is connected, for some $Y\subseteq X$ finite. The claim is intuitively obvious to me, but is not finding the ...
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39 views

Is the boundary of $ Q=\bigcup_{t \in (0,T)}\{t\}\times S(t) $ empty when $S(t)$ is a boundaryless hypersurface?

For each $t$, let $S(t)$ be a compact hypersurface in $\mathbb{R}^n$ with $\partial S(t) = \emptyset$. Consider $$ Q=\bigcup_{t \in (0,T)}\{t\}\times S(t) $$ Is the boundary of $Q$ empty? I don't ...
2
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1answer
55 views

Why would $[0,1) \times \eta$ (with lexicographic order topology) not be a manifold for $\eta > \omega_1$?

From Wikipedia's entry on the long line: And if we tried to glue together more than $\omega_1$ copies of $[0,1)$, the resulting space would no longer be locally homeomorphic to $\mathbb{R}$. ...
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1answer
121 views

Injective immersion (between smooth manifolds) that is no homeomorphism onto its image

Is there an injective immersion between smooth manifolds that is no homeomorphism onto its image? With smooth I mean $C^\infty$-manifolds and of course also the immersion should be $C^\infty$.
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1answer
65 views

Is $\lVert u \rVert_{L^2(M)} + \lVert \Delta u \rVert_{L^2(M)}$ equivalent to $\lVert u \rVert_{H^2(M)}$?

On a bounded Riemannian manifold without boundary, is it true that the norms $$\lVert u \rVert_{L^2(M)} + \lVert \Delta u \rVert_{L^2(M)}$$ is equivalent to the full $H^2$ norm $\lVert u ...
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0answers
56 views

Removing the boundary from a manifold with boundary

Let $M$ be a manifold with boundary $\partial M$. Can we say that $M'=M-\partial M$ is a closed manifold. I think it is correct, for example take $D$ to be the disk this a manifold with boundary the ...
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0answers
56 views

What are different geometric interpretations for the sample variance?

I primarily work with non Euclidean data and am looking to extend concepts of 'variance' to Riemannian manifolds. I am aware of Karcher variance, but I need efficient ways to solve for it. For ...
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1answer
52 views

pullback on vector fields and 1-forrms

On a Riemannian manifold $M$ one can identify 1-forms and vector fields by $$ \alpha(p) = \langle X(p),\cdot\rangle_p $$ Since we can perform a pullback on both 1-forms and vector fields I expected ...
2
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1answer
96 views

Why should the diffusivity matrix (of elliptic operator) map tangent space to itself?

I have seen that an elliptic operator $A$ on a hypersurface $\Gamma$, written as $$Au=-\nabla_\Gamma \cdot (M(x)\nabla_\Gamma u)$$ (where $\nabla_\Gamma$ is the tangential or surface gradient) is ...
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1answer
76 views

Few questions about global analysis relating $C^k$ functions

First question is about the definition. Let $U$ be an open subset of $R^n$. Let $f$ be $k$ times continuously differentiable function on $U$. $C^k$ norm of $f$ is defined as sum of uniform norm of ...
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1answer
51 views

Question related to tangent space of $U(n)$ at a matrix $g\in U(n)$

I was working on a homework problem that involved showing that the map $f:U(n)\rightarrow S^1,g\mapsto det(g)$ is a submersion (which is given here) And the following question emerged: Given $g\in ...
4
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2answers
291 views

Any N dimensional manifold as a boundary of some N+1 dimensional manifold?

Is this statement true: Question: Can any N dimensional manifold be realized as a boundary of some N+1 dimensional manifold? If so/not, how to prove/disprove it? I read a TQFT paper from Edward ...
2
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73 views

Nonvanishing vector fields on $\mathbf{S}^2$ and bases for $T_p\mathbf{S}^2$.

A vector field on a manifold is a (continuous, differentiable) map $X: M \to TM$ such that $X(p) \in T_pM$ for each $p \in M$. The tangent space $T_pM$ has a basis. Take $M = \mathbf{S}^2$. For each ...
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57 views

Showing that a map $h:S^2\rightarrow \mathbb{R}^4$ is an immersion

The Problem Let $h:S^2\rightarrow \mathbb{R}^4$ be a smooth map of the form $$ h(x,y,z)=(zy,yz,zx,ax^2+by^2).$$ Show that $h$ is an immersion for any $a,b\in \mathbb{R},a,b\neq 0,ab<0$. Attempt ...
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2answers
165 views

Unitary group and unit circle

Let $U(n)$ denote the group of complex unitary matrices, let $S^1$ be the unit circle in the complex plane. Then the map $$f:U(n)\to S^1,\quad f(A)=det(A)$$ is a group homomorphism and a submersion. ...
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2answers
64 views

Hessian of Morse function on $S^{n}$ mistake

I am trying to get that $f(x_{0},...,x_{n+1})=x_{n+1}$ has $index_{(0,...,0,1)}=n$ Can you find my mistake or post a partial solution? My attempt I evaluate df using inverse of stereographic ...
3
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1answer
214 views

Use of cut-off functions and partitions of unity

This is a simple problem, but I would still be very thankful if you could give me an advice on it. I'm trying to show that in a compact M-dimensional manifold, $$\int e^w \sqrt{g}\, dx \leq C \exp ...
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2answers
291 views

Finding a diffeomorphism between two smooth structures of $\Bbb R$

This is taking from Tu's Introduction to Manifolds book. We have defined $\mathbb{R}$ as the real line with the differentiable structure given by the maximal atlas of the chart ...
3
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2answers
110 views

Is this set a manifold?

For which $ ( \alpha , \beta ) \in \Bbb R^2$ set: $\{ (x_1,x_2,x_3,x_4) \in \Bbb R^4 | x_1+x_4= \alpha, x_1 x_4 - x_2x_3 = \beta \}$ is a manifold? I made a Jacobian matrix: $ \begin{bmatrix} ...
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1answer
231 views

Torus, manifolds

I have some trouble with the following questions: $\mathbb{R}^3$ has standard coördinates $(x, y, z)$. Regard in the plane $x=0$ the circle with centre $(x,y,z) = (0,0,b)$ and radius $a$, ...
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2answers
84 views

How to motivate vectors as derivations?

In a manifold it's easy to motivate the definition of vectors as equivalence classes of curves. On the other hand the definition as derivations is harder to motivate. I know how to show that the space ...
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1answer
78 views

Show that a set is a manifold.

Let $n \ge 3 $. How can I show that $M:= \{(x_1,...,x_n) \in \Bbb R^n \setminus \{(0,...,0)\} | x_1^2+...+x_n^2 = x_1 \cdot...\cdot x_n \}$ is a manifold of class $C^1$? Can anyone please tell me ...
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1answer
350 views

Munkres' Question on Manifolds

In Munkres' 'Analysis on Manifolds' on pg. 208 there's a question which reads: QUESTION: Let $f:\mathbb R^{n+k}\to \mathbb R^n$ be of class $\mathscr C^r$. Let $M$ be the set of all the points ...
2
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1answer
53 views

Intersection of two open sets in the projective plane

I want to compute the cohomology groups of the real projective plane, $P^2$, using Mayer Vietoris exact sequence. Now $H^0(P^2)=\mathbb{R}$, $H^2(P^2)=0$ being $P$ not orientable, so my problem really ...
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1answer
834 views

Why are de Rham cohomology and Cech cohomology of the constant sheaf the same

I am comfortable with de Rham cohomology, sheaves, sheaf cohomology and Cech cohomology. I am looking to prove the following theorem: If $M$ is a smooth manifold of dimension $m$, then we have ...
2
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1answer
74 views

vector field on $\mathbb{R} P^2$

Actually this is a quesion in Lee's book, Manifolds and differential geometry. I have problems working with projective spaces as manifolds.(e.g. what are curves in projective spaces ? I need to know ...
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1answer
97 views

Orientation of $X \times Y$

Suppose that $X$ is not orientable. How can I show that $X \times Y$ is never orientable, no matter what manifold $Y$ may be? I've tried supposing that $X \times Y$ is orientable, then using that ...
3
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1answer
93 views

Degree of polynomial seen as a smooth map

I need some help with a part of an exercise. Let $P$ be a real polynomial of degree $d$, seen as a map $P:\mathbb{R}\rightarrow\mathbb{R}$. Prove that if $d$ is even then the degree of $P$, $degP$, ...
3
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2answers
291 views

If a smooth map between manifolds is injective, is the induced map on the tangent spaces injective too?

If $\phi:M\longrightarrow N$ is an injective smooth map between two manifolds, then is $d\phi_m:M_m\longrightarrow N_{\phi(m)}$, the induced map between the tangent spaces injective too? I tried the ...
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1answer
44 views

Extension of funcion

I think its right but Im not sure. I have topological space (exactly manifold - second countable, Hausdorff, local Euclidean topological space) M, dim M=m. Let $A \subset M$ is closed set, dim A=n, ...
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1answer
50 views

Which integral curves of a field are defined for all times t?

Which integral curves of the field $X=x^2 \frac{\partial}{\partial x} + y \frac{\partial}{\partial y}$ are defined for all times t? I would be very thankful if somebody can help me understand what ...
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0answers
77 views

Any material on complexification?

These days, I met a problem on linear algebra: Suppose $A,B$ are real matrices. If there's a complex unitary matrix $U$ such that $U^*AU=B$, where $U^*=\overline U^\top$, namely, the conjugate ...
0
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1answer
46 views

Relative Compactness $\Rightarrow$ Compactness

I try to figure out: $(\overline{A}^U\text{ compact in }U )\Rightarrow( \overline{A}^X\text{ compact in }X)$ ...while $U\in\mathcal{T}$ It's clear for the case: $\overline{A}^X\subseteq U$ But else, ...
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1answer
300 views

Question about statement of Rank Theorem in Rudin

Theorem Suppose $m,n,r$ are nonnegative integers, $m\ge r, n\ge r$, $F$ is a $C^1$ mapping of an open set $E\subset \mathbb{R}^n$ into $\mathbb{R}^m$, and $F'(x)$ has rank $r$ for every $x\in E$. ...
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2answers
91 views

Distributional derivatives on hypersurface?

In a paper I was reading, the define a set $Q=(0,T)\times \Omega$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain, and then they write $$\langle \frac{d}{dt}u - \Delta u, \varphi ...