For questions about smooth manifolds.

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Sending unitary group to reals, second differential at critical points.

Let$$X = U(n) = \{B \in \text{GL}_n(\mathbb{C}): BB^* = I\}$$be the group of $n \times n$ unitary matrices over $\mathbb{C}$, viewed as a real manifold. Fix a diagonal $n \times n$ matrix $A$ with ...
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0answers
22 views

Generalization of Inverse Function Theorem to noncompact submanifolds

In Guillemin and Pollack's Differential Topology, they give as an exercise (#1.8.14) to prove the following generalization of the Inverse Function Theorem: Use a partition-of-unity technique to ...
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1answer
15 views

Show that the outward unit normal is smooth vector field

Exercise 2.1.8 of Guillemin and Pollack asks us to prove that if $X^m \subset \mathbb{R}^n$ is an embedded submanifold with boundary, then the outward unit normal to $\partial X$ is a smooth function ...
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39 views

Spivak's smooth partition of unity [duplicate]

You are right for your link But In your address, There is not any solution for this question and somebody had said that $f$ is redandant without that present even a reason or one proof or a rational ...
2
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0answers
24 views

Characterization of graphs of maps between smooth manifolds

Theorem 6.52 in Lee's Introduction to Smooth Manifolds, 2nd ed., says Suppose $M$ and $N$ are smoothe manifolds and $S \subset M \times N$ is an immersed submanifold. Let $\pi_M$ and $\pi_N$ ...
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37 views

Exponential map on a sphere in spherical coordinates

Let $M = \{ (x^\varphi, x^\theta) : x^\varphi \in [0, \pi), \thinspace x^\theta \in [0, 2\pi) \}$ be a manifold with metric $\mathrm{d}s^2 = (\sin x^\varphi)^2 (\mathrm{d} {x^\theta})^2 + (\mathrm{d} ...
2
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1answer
31 views

The meaning of “surjective” in the context of smooth manifolds

Nigel Hitchin, in a paper on differentiable manifolds (https://people.maths.ox.ac.uk/hitchin/hitchinnotes/manifolds2012.pdf), he states the theorem: Theorem 2.2 Let $F : U \rightarrow {\rm R}^m$ be ...
2
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2answers
40 views

Smooth self maps of compact manifolds.

Suppose $M$ is a compact $n$ dimensional manifold. Does there exist an example of the following: A smooth map $f: M \rightarrow M$ such that there exists $x \in M$ where $\mbox{d}_x f$ has maximal ...
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1answer
22 views

Notation in exterior algebra

I am following a course on introduction to manifolds by myself and I got stuck by a notation I don't understand. It defines the permutation group $S_n$, and then the signature of a permutation as ...
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2answers
43 views

2-form on a smooth manifold

Let $M$ be a smooth manifold, $f:M$ $\rightarrow \mathbb{R}$ differentiable and $p\in M$ with $df(p)=0$. I am trying to show that the application, $$\begin{matrix}\mathfrak{X}(M)\times ...
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0answers
19 views

Charts in an oriented manifold with boundary

Let $M$ be an oriented manifold (with boundary) with $dim (M)\ge 2$. Show that there exists an atlas $\{(U_{\alpha},\phi_\alpha)\}_{\alpha\in I}$ for the chosen orientation such that ...
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2answers
124 views

What hyperbolic space *really* looks like

There are several models of hyperbolic space that are embedded in Euclidean space. For example, the following image depicts the Beltrami-Klein model of a hyperbolic plane: where geodesics are ...
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3answers
206 views

How can one compare these two 4-manifolds

We would like to compare the following two real 4 dimensional manifolds: 1)$M$=The tangent bundle of $S^{2}$ 2)$N$= The total space of the canonical line bundle over $\mathbb{C}P^{1}\simeq S^{2}$ ...
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0answers
41 views

an isomorphism between the tangent space of a manifold to euclidean space

1) I was told in class many years ago that, the tangent space if the sphere $\mathbb{S}^2$at a point $p$, i.e. $T_p\mathbb{S}^2$ is isomorphic to $\mathbb{R}^2$. Could anyone give me a proof of ...
5
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1answer
75 views

Diffeomorphism group of a manifold is never a Lie group?

Let $M$ be a smooth manifold. I heard there is a way to introduce a topology and a structure of infinite dimensional manifold (something like a Banach or a Frechet manifold) on $\text{diff}(M)$ ...
2
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0answers
13 views

Solutions to Dirichlet problem on manifolds with boundary

I am looking for a reference for the following assertion: Let $M$ be a Riemannian manifold with boundary, and $f:\partial M \rightarrow \mathbb{R}$ be smooth. Then there exists a unique smooth ...
3
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0answers
74 views

Equivalent condition for non-orientability of a manifold

I've just came across this question, which gives us a great tool for showing that smooth manifold is non-orientable. Namely Thm. If $M$ is a smooth manifold and there are two charts ...
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1answer
37 views

Generalizations of Inverse Function Theorem

A beautiful exercise in Guilleman and Pollack asks us to show the following generalization of the Inverse Function Theorem: Suppose $f: M \to N$ is a map of smooth manifolds, and $Z$ is a ...
0
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1answer
27 views

Analytic expression of a 1-form

Let $M$ be a differentiable manifold, $V\in\mathcal{X}(M)$ a vector field on $M$ and $\alpha\in\mathcal{X}^*(M)$ a 1-form. Let $L_{V\alpha}$ be another 1-form defined by: ...
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0answers
37 views

Reference request: Pullbacks along submersions are submanifolds and the induced map is a submersion

I'm looking for an introductory book in differential topology in which there are proofs of the following facts: Let $X,X',Y$ be smooth manifolds, $X\rightarrow Y$ a smooth map and $X'\rightarrow Y$ ...
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0answers
26 views

Lie group is parallelizable

While going through the proof given by Alex Youcis at http://math.stackexchange.com/a/308798/86801 , I found the part where the local representation of the map $\ \Phi\ $ is shown to be the identity ...
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1answer
31 views

What do tensors of second order map to?

On page 15 of James G. Simmonds book "A brief on Tensor Analysis" (chapter 1 of the first published edition), a second order tensor is described as an operator that sends vectors into vectors. On ...
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1answer
28 views

Preimage Lemma for transverse map. Help with some passages

I'm on my way proving the Preimage Lemma for a transverse smooth map but I've encountered some problems with two passages: Let $f\colon M \to N$ be a smooth map transverse to a submanifold $L$ of ...
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0answers
34 views

Relation between integral curves

Let $M$ be a manifold, $f:M\rightarrow\mathbb{R}$ a differentiable map and $X$ a vector field on $M$. I'm trying to find a relation between the integral curves of $X$ and $e^fX$. I am not quite sure ...
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2answers
40 views

Volume forms and volume of a smooth manifold

Choose a volume form $\omega$ on $M$, oriented manifold. For every $F\in C^{\infty}_c(M)$, we define $$ \int_M F:=\int_M F\omega $$ where in the right hand term $M$ is taken wit positive orientation ...
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0answers
27 views

Vector fields on a manifold

Let $M$ be an arbitrary manifold, $p\in M$ and $0\neq\xi\in T_pM$. I have to proof there are vector fields $V$ and $W$ in $M$ with $V(p)=W(p)=\xi$ but $[V,W]|_p\neq 0$. If you choose a chart ...
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2answers
49 views

Is this a correct way of thinking about diffeomorphic manifolds?

In set theory there is the concept of a bijection, a one-to-one correspondence between the elements of $2$ sets. In topology the concept of a homeomorphism $f:X\to Y$ is quite easy to wrap your head ...
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0answers
32 views

continuous (smooth) maps and group homomorphism

Consider a topological group $G$ (or smooth Lie group) and a topological space $M$ (or smooth manifold) and a group homomorphism $\phi:G\rightarrow Sym(M)$, where $Sym(M)$ is the symmetry group of M, ...
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1answer
23 views

Proof that a function $f:\mathbb{R}\times\mathbb{R}\to N$ restricts to a smooth function on $S^1$

I have to prove Proof that a function $f:\mathbb{R}^2\to N$ (where $N$ is a smooth manifold) restricts to a smooth function on $S^1$ Here $S^1$ is defined as the subset of $\mathbb{R}^2$ ...
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0answers
30 views

A detail in the proof orientable manifold admits exactly two orientations

Let $M$ be a real manifold and let $\{(U_\alpha,\phi_{\alpha})\}_{\alpha\in I}$ $\{(V_\beta,\psi_\beta)\}_{\beta\in J}$ be two oriented atlases. Let's define, for $p\in U_\alpha\cap V_\beta$, ...
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1answer
17 views

Local extension of a function on an immersed submanifold

Consider the following passage in Spivak's Differential Geometry book: I am having trouble understanding where he says $g = \tilde{g} \circ i$ on $V \cap M_1$. Since $V$ is (I think) supposed to be ...
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1answer
269 views

Can the interior of a manifold be orientable but not its boundary?

Suppose $M^m$ is a manifold with boundary. If we are given an orientation for $M$, we can then derive an orientation for $\partial M$ by considering the orientation of $TM$ at $\partial M$ and then ...
0
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1answer
32 views

Lie bracket in coordinates

In $\mathbb{R}^2-\{0\}$ we consider the vector field defined by, $$V=-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y}$$ I am trying to find all other vector fields $X$ that $[V,X]=0$. My ...
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1answer
31 views

Existence a diffeomorphism on $\mathbb R^n$

How may I show that for any $p,q\in\mathbb R$, there exist a diffeomorphism $F:\mathbb R\rightarrow\mathbb R$ such that $F(p)=q$ and $F$ to be an identity function outside of a some neighborhood of ...
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48 views

Closed orientable three manifold with finite cover by $S^1 \times S^2$ or $T^3$

I have been thinking about a problem where I can conclude that I have a closed orientable three manifold which is covered by $S^1 \times S^2$ or $S^1 \times T^2$. I think that the geometrization ...
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0answers
28 views

Derivatives of the reciprocal of a smooth function

I am trying to find a smooth function f(t) such that its n-th derivatives are bounded by the n-th derivatives of $Ce^{Ct}$, $\forall n \in N$ and the n-th derivatives of its reciprocal are ...
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87 views

Newton iteration on Riemannian manifolds

Suppose $f:M \to N$ is a smooth map between complete Riemannian manifolds of the same dimension. Suppose $Df(m_0)$ is invertible, and $n$ is a point close to $f(m_0)$. Can we perform Newton iteration ...
2
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2answers
43 views

Compute the tangent space at the unit matrix

Compute the tangent space $T_pM$ of the unit matrix $p=I$ when $$(i)\,M=SO(n)\\ (ii)\,M=GL(n)\\ (iii)\,M=SL(n).$$ My attempt: I think I have computed the tangent space in the case that $M=SL(n)$. ...
2
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1answer
29 views

Given local smooth extensions, construct a global smooth extension

In Spivak's A Comprehensive Introduction to Differential Geometry, Vol. 1, he defines a function from a half-space $H^n$ to be $C^\infty$ if there is an extension to a neighborhood of $H^n$ that is ...
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1answer
23 views

Coordinates on manifold and tangent space

Let $M$ be smooth manifold and $x \in M$. $\langle v_1,\dots,v_n \rangle = T_xM$ i want to find chart $(U,x)$ such $v_i = \frac{\partial }{\partial x_i}$. Ok there is some chart $(W, y)$ and we have ...
0
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1answer
31 views

A question about an explanation of the image of an immersion f:X$\to$Y need not to be a submanifold of Y

The explanation is: From the Local Immersion Theorem,it is evident that f maps any sufficiently small NBHD W of an arbitrary point x diffeomorphically onto its image f(W) in Y. So every point in the ...
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1answer
32 views

Metric Isometry is always smooth?

Let $M$ be a smooth manifold. Let $d$ be any metric on $M$ which induces the topology on $M$. Let $f:(M,d) \rightarrow (M,d) $ be an isometry (in the sense of metric spaces). Is it true that $f$ must ...
0
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1answer
24 views

Construction of partitions of unity in Warner

On p. 11 of Warner's Foundations of Differntiable Manifolds and Lie Groups, he discusses partitions of unity. The theorem says Let $M$ be a differentiable manifold and $\{U_\alpha: \alpha \in A ...
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Counting the number of connected components in a complement

I have come across the following problem. Suppose you are given a closed, connected, orientable n-dimensional submanifold without boundary, in $\mathbb{R}^{n+1}$, call this submanifold $K$. Prove that ...
2
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1answer
32 views

Proving this result on tangent spaces to foliations

Reading through Lee's introduction to smooth manifold, I bumped into this result: I've tried to prove it, but have gotten stuck. A foliation is basically slicing $M$ into $k$-dimensional ...
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1answer
108 views

Two nonevident implications in a proof

I am reading part of Lee's introduction to mainfolds. I got to the following proposition. I am having trouble between the two displayed lines of the proof. Precisely, my questions are: How does ...
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0answers
31 views

Bump function which $=1$ exactly on some compact set

Given an open set $O \subset \mathbb{R}^n$, and a compact set $K \subset O$, there exists a $C^\infty$ function $\varphi: \mathbb{R}^n \to [0,1]$ which is constantly $1$ on $K$ and is zero outside ...
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1answer
37 views

Extending a smooth function of constant rank

Let's denote $\mathbb{H}^m = \{(x_1, \ldots, x_m) \in \mathbb{R}^m\ |\ x_m \geq 0\}$. For an open subset $U \subset \mathbb{H}^m$, a function $f : U \to \mathbb{R}^n$ is called smooth if it can be ...
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Ramanan's definition of differentiable function

In his book Global Calculus, Ramanan defines a differential manifold as follows: What is meant by condition (b)? Is $\mathcal A$ simply a subsheaf of the sheaf of real valued continuous functions ...
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Zeroes of differential form embedded

Let us consider $S^1$ as a manifold embedded in $\mathbb{R}^2$. Let $dx_1\in\Omega^1(\mathbb{R^2})$. $$ Z_{\mathbb{R}^2}:=\{p\in\mathbb{R}^2:(dx_1)_p=0\}\\ ...