Questions tagged [smooth-manifolds]

For questions about smooth manifolds, a topological manifold with a maximal smooth atlas.

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21 views

Smoothness of a homomorphism on Lie subgroup

Suppose $\phi:G_1\to G_2$ is a Lie group homomorphism where $G_1$ is connected. Let $\phi(G_1)\subseteq G_3$ and $G_3$ is a Lie subgroup of $G_2.$ I want to prove that $\phi:G_1\to G_3$ is a Lie group ...
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27 views

How to obtain real vector from abstract tangent vector in the case of the manifold $\mathbb R^n$

I know that for every $p\in\mathbb{R}^n$ the map \begin{align} \Phi_p\colon\mathbb{R}^n&\to T_p\mathbb{R}^n\\ v&\mapsto D_{v,p} \end{align} is an isomorphism, where \begin{align} D_{v,p}\...
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8 views

Wave map on manifolds

Let $u: V \rightarrow M$, where $(V,g)$ is a Lorentzian manifold and $(M,h)$ is a Riemannian manifold. Also $V=S \times R$, where $S$ is an $n$-dimensional orientable smooth manifold. Now in a paper ...
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1answer
17 views

Show that $C^\infty(\overline{\Omega}) \subseteq C^{0,1} (\overline{\Omega})$

Let $\Omega$ be a bounded, connected, open domain in $\mathbb{R}^d$ with smooth boundary. Denote by $C^{0,1} (\overline{\Omega})$ the space of continuous functions $u$ on $\overline{\Omega}$ such ...
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8 views

Structure preserving maps between manifolds with boundary

While learning the basics about smooth manifolds with boundary in this semesters' course about analysis on manifolds, there's a seemingly basic property I didn't find anywhere. Namely I want to ...
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1answer
26 views

Confusion about the Definition of Smooth Functions on a Manifold

I am slightly confused about the definition of smooth functions on a smooth manifold given in An Introduction to Manifolds by Loring Tu (Second Edition, page no. 59). The definition is given below. I ...
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1answer
38 views

Interesting examples of submersions that are not surjective

What are some "interesting" examples of submersions that are not surjective? Usually, the notion of submersion comes with a prefix of surjective. Most of the maps we come across when we do ...
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2answers
41 views

Definition of Poisson Bracket

Context: Let $f,g :T^*M\rightarrow \mathbb{R}$, the Poisson Bracket was defined classically as $$\{f,g\}=\sum\limits_{i=1}^n\frac{\partial f}{\partial q^i}\frac{\partial g}{\partial p_i}-\frac{\...
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28 views

Topological decomposition of a smooth manifold

The decomposition of a smooth manifold for topological analysis requires that all vertices lie at the ends of edges, all edges bound the incident faces, and so on. Thus for example a decomposition of ...
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29 views

Prerequisite for Milnor Toplogoy from a Differential Viewpoint [duplicate]

I finished Introduction to Manifold by Loring Tu and I am interested to study this book by Milnor. But I didn't take a course in topology (other than the one I learnt from the appendix of Tu's book). ...
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34 views

Prerequisite for Milnor Topology from a Differential Viewpoint

I finished Introduction to Manifold by Loring Tu and I am interested to study this book by Milnor. But I didn't take a course in topology (other than the one I learnt from the appendix of Tu's book). ...
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32 views

Canonical one-form: Why is it called that when it maps vectors to vectors? Can dual basis at some point act on a tangent space at a different point?

Let $G$ be a Lie group and $T_g G$, $T^*_g G$ denote the tangent and cotangent spaces at a point $g \in G$, respectively. Let $e$ be the unit element and $X_V$ the left-invariant vector field ...
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1answer
56 views

Show that there is a $\pi_i$-related smooth vector field for each smooth vector field $X_i \in \Gamma(M_i,TM_i)$

Assume $M_1, \dots,M_k$ are smooth manifolds and define $M:=M_1\times \dots \times M_k$. Denote the projections on the $i$-th factor with $\pi_i: M \rightarrow M_i$. I want to show that for each ...
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1answer
58 views

A new type of manifold, is such a construction interesting? Is it relevant for the Euler-Lagrange equations

Recently, I've been wondering how to rewrite the standard Euler-Lagrange equations: \begin{align} \dfrac{\partial L}{\partial q^i} - \dfrac{d}{dt} \left(\dfrac{\partial L}{\partial \dot{q}^i}\right) &...
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42 views

Topological Manifold which does not admit Riemann surface structure?

I have a question motivated by some examples in section 4.2 of Schlag's A Course in Complex Analysis and Riemann Surfaces. Example 1. Any smooth, orientable two-dimensional submanifold of $\mathbb{R}...
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21 views

Coordinate curve depends on $t$

I have a basic problem to understand the coordinates $(x)$ on a manifold $M$. The coordinate vector is $$\frac{\partial}{\partial x^j}$$ which says that the tangent vector depends on $x^j$ but the ...
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1answer
23 views

Coordinate vector and curve

How can I see that the $j$-th coordinate vector $$\frac{\partial}{\partial x^j}$$ on a manifold is the velocity vector to the $j$-th coordinate curve parametrized by $x^j$. Both, by velocity and ...
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41 views

Silly doubt on differential forms notation

A little fact is confusing me concerning Differential forms. Besides all kinds of motivation, a differential $p-$form is just the following tensor: $$\textbf{F} = F_{\mu_{1}...\mu_{p}}(x^{\nu})\...
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18 views

Is there a definition of limiting flatness on a manifold?

I am not talking about local flatness. Local flatness means an entire chart can be mapped to a flat euclidean space. For this reason for example a sphere is not flat, because any chart will still be ...
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26 views

Hint in an exercise, manifold and its dimension

I'm triynig to solve this exercicse but I have a trouble: I don't know what is the dimension of $G$, it is finite or not? but I suppose the base is $(x_1,...x_n)$ so I can wrote any vector as follows ...
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34 views

Connections and second derivatives of curves

If $M$ is a smooth manifold, let $AM$ be the subspace of the double tangent bundle $TTM$ consisting of vectors $v$ such that $\pi^{TM}(v) = \pi^X_*(v)$, where $\pi^X: TX \to X$ is the projection of a ...
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1answer
29 views

construct nonvanishing one form on $\mathbb{RP}^3$

I met such a problem in my homework: construct an everywhere nonvanishing 1-form on $\mathbb{RP}^3$; can your construction be generalized to $\mathbb{RP}^n$? I know nothing about the background of ...
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10 views

Binary representation unusual relation in theorem about immersed manifolds.

Question I'm reading Schuller's Lectures on the Geometric Anatomy of Theoretical Physics and he states the following theorem. I was surprised by this theorem and would like references for further ...
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1answer
37 views

Wrong proof of $TM$ is diffeomorphic to $M\times \mathbb{R^m}$

I want to see what's wrong in here: Let $M$ be a smooth manifold with dimension $m$. I will show $TM$ is diffeomorphic to $M\times \mathbb{R^m}$. proof) Define $F:TM\rightarrow M\times \mathbb{R^m}$ ...
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31 views

Ring of smooth functions proof

$C^{\infty}(U)$ is a ring. Where $U$ is open in $\mathbb{R}^n$. Not $D_jf(a)$ denote the partial derivative of $f$ in the direction of $e_j$ at a. My attempt: Abelian group: Let $f,g\in C^{\infty}(...
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1answer
20 views

lie bracket is vanish iff translation is commute

I want to prove following claim: Given two vector fields $X$ and $Y$ on smooth manifold $M$, $[X,Y]=0$ if and only if $\Phi^X_t \circ \Phi^Y_s \circ \Phi^X_{-t} \circ \Phi^Y_{-s}(q)=q$ for $\forall ...
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1answer
34 views

Tangent space identification basis

Let $e_{i_p}$ denote the standard basis for $T_p(\mathbb{R}^n)$. Theres a vector space isomorphism between $T_p(\mathbb{R}^n)$ and $D_p(\mathbb{R}^n)$, where $D_p$ is the set of derivations at $p$, ...
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47 views

Show that S is an embedded $k$-submanifold of $M$

Assume $M$ is a smooth manifold and $S$ is a subset of $M$ such that each point $p\in S$ has a neighborhood $U \subseteq M$ such that $S\cap U$ is an embedded $k$-submanifold of $U$. I want to show ...
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33 views

Exact one form on non-simply-connected manifold

Let $M$ be a smooth $n-$dimensional manifold and $\alpha\in\Lambda^1(M)$ a smooth $1-$form. Let $N\subset M$ be an embedded submanifold where $\alpha|_N=0$. If $d\alpha=0$ on the whole $M$, can I say ...
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1answer
66 views

Why do we use tangent bundles to define vector fields on manifolds?

Most textbooks define a vector field on a smooth manifold $M$ as a section of the tangent bundle of $M$. My question is: why is it even necessary to talk about bundles when defining vector fields on $...
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1answer
50 views

Understanding Hessian on Manifold (without Riemannian Geometry)

I've been going through notes on Morse theory and Handlebody theory and I've been having some trouble with the definition of the Hessian provided. The notes are on pages 3-4 here http://people.math....
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19 views

lower bound the distance between two varieties

$\DeclareMathOperator{\complex}{\mathbb{C}}$ Let $X,Y \subseteq \complex^n$ be homogeneous, smooth, irreducible, closed algebraic sets with $X \cap Y=\{0\}$. I would like to numerically lower bound ...
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37 views

Coordinate curves and the Lie bracket on a differentiable manifold

On a differentiable manifold, let $x^i$ be the coordinates of points on the manifold in a local parametrization. Then it is straightforward to show that the Lie bracket between vector fields defined ...
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33 views

How to calculate the Riemannian gradient in practice (optimization)?

In a couple of papers, I saw steepest descent defined by $\mathbf{v} = -\mathbf{G}_p^{-1}\nabla f(\mathbf{p}) \in T_x\mathbb{R}^n$ where $\mathbf{G}$ is the Riemannian metric tensor of an manifold. ...
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29 views

Attempt to give an alternate definition of Cotangent Space in terms of integration

Let $M$ be the manifold, then we know that tangents space has basis $\frac{\partial}{\partial{x_i}}$ and it acts on $g:M \rightarrow \mathbb{R}$ as $\frac{\partial{g}}{\partial{x_i}}$ and cotangent ...
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14 views

The same generalized Gauss map

Let $f:M^{n}\rightarrow \mathbb{R}^{m}$ an isometric immersion of a simply connected Riemannian manifold such that $\Phi\in \Gamma(\mbox{End}(TM))$ is a Codazzi tensor on $M^{n}$, that is, $$(\nabla_{...
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1answer
33 views

how to deduce that this concrete differential form isn't exact?

I met such an exercise in my smooth-manifolds course, which made me very much puzzled. Let $U=\mathbb{R}^{n} \backslash\{0\}$ and $\omega=\sum_{i=1}^{n}(-1)^{i+1} f_{i} \mathrm{d} x^{1} \wedge \ldots \...
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1answer
70 views

Second derivative test (and sign of laplacian at critical points) for manifolds

I'm trying to understand in more detail some of the justifications for a proof of the second derivative test for Riemannian manifolds, given below: I've never seen the Laplacian interpreted as an ...
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1answer
57 views

Equivalent definitions of the Möbius band

In Manfredo do Carmo's Riemannian Geometry, the Möbius band is defined as the quotient of the cyllinder $S = \{ (x, y, z) \in \Bbb{R}^3 \ : \ x^2 + y^2 = 1, \ - 1 < z < 1\}$ by the group $\{A, ...
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12 views

existence of a transverse homotopic function - example

I would like to find an example for the following theorem: Let $f: M \to N$ be a smooth map. There exists a map arbitrarily close to $f$, and homotopic to it, which is transverse to $Z$. In the ...
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53 views

A confusion about flows

First of all I apologize for asking this basic question. Let $M$ be a smooth manifold with diffeomorphism group $Diff(M)$. My understanding of a (smooth) flow on $M$ is a group homomorphism $\mathbb{R}...
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60 views

Area convergence

Let $(M^3,g)$ be a compact, connected and oriented Riemannian manifold with boundary. Let $\{\Sigma_n\}_{n \geq 1}$ be a sequence of free boundary minimal surfaces embedded in $M$ that converges ...
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3 views

How are two embeddings of orbifolds are related?

The following Lemma is used to prove that smooth embedding of orbifold charts give rise to injective homomorphism in groups acting on respective manifolds, how do I prove following: Given 2 embeddings ...
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1answer
40 views

Guillemin and Pollack Exercise 4.4.5

A closed curve $\gamma$ in a manifold $X$ is defined to be a smooth map $\gamma: S^1 \to X$, and I am tasked with finding an explicit formula for the line integral $\oint_{\gamma}\omega = \int_{S^1} \...
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14 views

collection of $m-$frame is a Manifold (Stiefel manifold)

Let $X = \mathbb{R}^{m+n}$ be the space. Define $$F_m(X) = \{(v_1, v_2, ..., v_m) : v_i \in X, \{v_1, v_2, ..., v_m\} \ \mbox{is linearly independent}\} \subseteq \mathbb{R}^{m+n} \times ... \times \...
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1answer
52 views

Equation with differential forms

I am interested in finding conditions to impose over a one form $\alpha$ defined over a torus $\mathbb{T}^2$ to be sure that there is no function $g:\mathbb{T}^2\rightarrow\mathbb{R}$ solving the ...
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0answers
47 views

Euler equation in a Riemannian manifold

I am trying to prove that the Euler equation for a Riemannian 3-manifold $M$ $$ \frac{\partial u}{\partial t} + \nabla_{u} u = - \operatorname{grad} p \qquad \text (1) $$ is equivalent to $$ \frac{\...
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1answer
151 views

Are Christoffel symbols associated with a tensor object?

First of all, my question lies on: Differentiable, real, n-dimensional Manifolds and in the context of differential geometry for General Relativity. Also, my level of academic mathematical language do ...
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1answer
46 views

Weakly irreducible manifold

Let $M^3$ be a compact, connected and orientable manifold with boundary. I will say that $M$ is weakly irreducible if every smoothly embedded $2$-sphere $S \subset \operatorname{int}(M)$ separates $M$,...
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1answer
26 views

Tangent space of the image of an immersion

I'm going to phrase my question in a certain context (while I do believe generalisations are possible!). Say we have an injectieve Lie group homomorphism $F\colon G\to H$ between Lie groups $G$ and $H$...

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