Questions tagged [smooth-manifolds]
For questions about smooth manifolds, a topological manifold with a maximal smooth atlas.
5,974
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Local structure of symplectic manifold
Algebraic geometry, in a naive setting, could be described as the study of spaces that locally are the solutions of systems of polynomial equations.
Similarly, locally any smooth manifolds can be ...
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Question about the definition of a vector field on a manifold
I am reading do Carmo's Riemannian Geometry, and he defines the tangent bundle $TM$ of a differentiable manifold $M$ as $\{ (p, v) : p \in M, v \in T_pM\}.$ That's fine. Next, he defines (on page 25):
...
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Proposition 1.1.14 D.J Saunders on Bundle
everyone.
I am studying D.J Saunders's book The Geometry of Jet Bundles. On proposition 1.1.14, a proof is given that the structure of the total space $E$ of a bundle $(E,\pi,M)$ depends on those of ...
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questions about regular level set theorem and manifolds
Regular level set theorem says that every regular level set of a smooth map is a closed embedded submanifold whose codimension is equal to the dimension of the range.
Is it if and only if?
For example ...
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Show that the product of any number of spheres can be embedded in some Euclidean space with codimension one
This problem is already solved in an elegant way here:
Product of spheres embeds in Euclidean space of 1 dimension higher
But I was trying to use a different approach:
I'm using induction, it's clear ...
- 171
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Computing the differential in local coordinates
Let $F:M\to N$ be a smooth map between manifolds and $p\in M$. I am having trouble understanding the differential $dF_p$. I understand how $\left\{\frac{\partial}{\partial x^1},..., \frac{\partial}{\...
- 189
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Subharmonic in a Neighbourhood of a critical point
$M$ is a compact Riemannian manifold and $f$ is a smooth function with the property that for each critical point there is a neighborhood in which $f$ is subharmonic. Can we say that $f$ is subharmonic ...
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Prove the flow of the left-invariant vector field $w_X$ at time $t$ is$\Phi_X^t(g)=g\exp (tX), t \in \mathbb R, g \in G$
If $M$ is a smooth manifold, $G$ a Lie group and $v, w$ are vector fields. For small $t \in \mathbb R$ there is a flow determined by $v$, $\phi^t:M \rightarrow M$ which is a diffeomorphism. The ...
- 2,696
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Show that the lens space is a smooth 3-manifold
Suppose we view $S^3\subset C^2$. Then for coprime integers $p,q$ we define the lens space by $M_{p,q}=S^3/\sim$ where $(z_1,z_2)\sim(z_1e^\frac{2\pi i}{p},z_2e^\frac{2\pi iq}{p})$. I want to show ...
- 189
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Understanding 3-manifold retraction to graph
I am reading On Fibering Certain 3-Manifolds by Stallings. It's a short paper, and I think I understand at a high level what's happening, but there are a few technical details I don't quite get. This ...
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Lots of questions about smooth manifolds and differentials
I just recently started differential topology and I am really struggling with some of the ideas. In particular, once we start talking about tangent vectors. Let $M$ be a n-manifold, then a linear map $...
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What does "mod" mean in the context of this tangent space proof in Warner?
I'm trying to read Frank Warner's Foundations of Differentiable Manifolds and Lie Groups and got confused with Theorem $1.17$ as some who does not have a pure mathematics background.
Let $F_m$ be the ...
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What is known about pairs of "physical" SDEs and "statistical" SDEs?
Background:
Recall that a Langevin motion on a Riemannian manifold $(M, g)$ in $\mathbb{R}^D$ can be written down as the solution to the SDE in a local chart $U\subset \mathbb{R}^d$ (open)
$$dY_t = [\...
- 1,247
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About local coordinate systems and the complete description of a manifold
I'm a physics student and most of the mathematics that I think I know I learned self-taught. That said, I have the following question that may be very naive but I don't know where to really ask.
If I ...
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Mathematical representation of projection plane
I've gotten the following assignment, given this specification of a screen I need to project a video onto, I want the video to appear flat.
My idea was to find a parametric equation describing this ...
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About $R$-algebras of smooth functions over manifolds [closed]
I am confused about finding a manifold whose $R$-algebra of smooth functions is a determined $R$-algebra.
smooth manifolds and observables jet nestruev
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Smoothness and its relation to infinte differentiability
Why does a function having any 'kinkiness' not infinitely differentiable?
Some of the functions(e.g. $f(x) = (|x|)^{k}$, $k \in \mathbb{N}$) belong to the class of $C^{k-1}$ functions(i.e. for these ...
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Why a vector field on a manifold defines a 1-1 mapping on the manifold?
I am reading paragraph 3.1 (Introduction: how a vector field maps a manifold into itself) of chapter 3 (LIE DERIVATIVES AND LIE GROUPS) of the book Geometrical Methods of Mathematical Physics, by B. ...
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Partitions of unity exercise: Showing existence of smooth function zero on closed subset.
Let $M$ be a smooth manifold. Let $B \subset M$ be closed and suppose $\delta:M \to \Bbb{R}^+$ be continuous.
(a) Using partitions of unity, prove there exists a smooth function $\tilde{\delta}: M \to ...
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There are smooth vector fields along a curve forming a basis of the tangent space at each point
An application of parallel transport is to prove the existence of smooth frame along a curve on the manifold. But it seems that the existence has nothing to do with the metric, so I would like to find ...
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Incompatible coordinate charts on the two dimensional sphere
Most of the examples of the charts for $S^2$ involve compatible ones like:
Stereographic projections from the North and South Pole.
Projections onto the disk bound by the equator.
What would be ...
- 153
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Tensor fields and scalar function pullbacks
For convenience, $(p,q)$ tensor fields on a differentiable manifold $M$ is defined to be the entire scalar field.
On the other hand, in my textbook, the pullback of the $(p,q)$ tensor $T(x)$ at $x \...
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smoothness of map from general linear group to symmetric matrices (manifolds)
Let $GL(n,R)$ be general linear group and $S(n,R)$ be set of symmetric matrices. Both viewed as manifolds.
Let $f: GL(n,R)->S(n,R), f(A)=A^TA$
I want to prove that $f$ is smooth map. How do I do it?...
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Restriction of Differential Form On a Connected and Compact Submanifold
Show that, conversely, if $M \subset \mathbb{R}^3$ is a compact and connected submanifold with the proper
$$
\left.(x d x+y d y+z d z)\right|_M=0,
$$
then $M$ is one of the spheres centered at the ...
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How to characterize the tangent space $T_f C^\infty(K, \mathbb{R}^n)$ and paths in $C^\infty(K, \mathbb{R}^n)$
Let $K \subset \mathbb{R}$ be compact. For any function $f \in C^\infty(K, \mathbb{R}^n)$ how would one characterize the tangent space $T_f C^\infty(K, \mathbb{R}^n)$?
I am following a set of notes ...
- 4,297
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If the image of an open set by a smooth map f is an embedded submanifold, is f a local embedding
My question is about the use of the "local slice criterion" (as presented in Lee's "Introduction to Smooth Manifolds") to obtain an embedded submanifold from some subset $S$ of a ...
- 577
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Smooth manifold with finite atlas
There is already quite a bit of discussion about this online, but I feel like a lot of this discussion is very confused so please help me understand the following questions:
Does every connected ...
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How can partial derivatives of a function be expressed as components of a covector field independently of coordinates?
At the end of page 280 of Lee's Introduction to Smooth Manifold the author states
Although the partial derivatives of a smooth function cannot be interpreted in a
coordinate-independent way as the ...
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A couple of questions regarding the Vector Bundle Chart Lemma
I have a couple of questions regarding the proof of the Vector Bundle Chart Lemma in Lee's Introduction to Smooth Manifolds:
Why is $\mathbb{H}^n\times\mathbb{R}^k\cong\mathbb{R}^{n+k}$, and what is ...
- 4,708
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Handles have the form $D^λ×D^{m−λ}$
I'm studying Matsumoto's An Introduction to Morse Theory. I want to solve a problem on page 76.
Context: Let $M$ be a closed $m-$manifold and $f:M\rightarrow \mathbb{R}$ a Morse function. Let $c$ be a ...
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Homogeneous space with finite invariant measure
Let $G$ be a locally compact second-countable Hausdorff group with a compact subgroup $H$. Assume that the modular functions of $G$ and $H$ agree on $H$. Therefore, there is a non-trivial left-...
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John Lee's ISM Problem 5-23 (smooth structure of regular level sets for manifolds with boundary)
The problem statement:
Suppose $M$ is a smooth manifold with boundary, $N$ is a smooth manifold, and $F: M \to N$ is a smooth map. Let $S = F^{-1}(c)$, where $c \in N$ is a regular value for both $F$ ...
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Proof of length of a tangent vector of a geodesic is constant on Finsler Manifold
In proof of Proposition 2.2 of the book Introduction to Modern Finsler Geometry by Shen and Shen I have faced the following problem:
Proposition 2.2: If $\sigma(t)$ is a geodesic on a Finsler manifold ...
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The relationship of tangent space between submanifold and manifold.
I was confused when I prove the next Proposition appeared in the book Introduction to smooth manifolds by Lee.
Suppose $M$ is a smooth manifold with or without boundary ,$S\subset M$ is an immersed ...
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Can the gluing lemma also hold for smooth maps that agree on closed sets?
In topology, the Gluing Lemma states that two continuous functions $f:A\rightarrow X$ and $g:B\rightarrow X$ defined on closed subsets $A$ and $B$ will define a continuous function on the union $A\cup ...
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Motivation for Lee's definition of smooth map
I am trying to understand why Lee makes the following definition for real valued smooth maps on manifolds.
Let M be a topological manifold. We call $f:M\to \mathbb{R}$ smooth if chart for $p \in M$, ...
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Hodge conjecture [closed]
Hello I'm trying to understand the idea behind Hodge conjecture and I have naive approach but what does it mean Hodge classes statement:
$H^{2k}(X,\mathbb{Q}) \cap H^{k,k}(X)$? these symbols? My ...
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Show that $X$ admits a natural almost complex structure and that any almost complex structure on $X$ is induced by a complex structure.
Let $X$ be an oriented Riemann surface. Show that $X$ admits a natural almost complex structure and that any almost complex structure on $X$ is induced by a complex structure.
Okay we want to define $...
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Prove that $\{g_{\alpha\beta}\}$ is a cocycle on $M$ whose associated vector bundle is the tangent bundle $TM$.
Let $\{(U_\alpha, \varphi_\alpha)\}$ be an atlas on a smooth manifold $M$ where $\varphi_\alpha:U_\alpha \to \mathbb{R}^n$, $\varphi_\alpha=(x^1_\alpha,\dots,x^n_\alpha), n = \dim M$. Let $g_{\alpha\...
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Is every smooth atlas for a manifold with boundary contained in a unique maximal smooth atlas?
I'm wondering if the following proposition (from Lee's Introduction to Smooth Manifolds) also applies to smooth manifolds with boundary:
Definition: a map from an arbitrary subset $A\subseteq \mathbb{...
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Is the total space of the frame bundle of a smooth manifold $M$ always spin?
I seem to remember reading in Kobayashi that the total space of a frame bundle is always parallelizable. If I'm remembering this correctly then the total space of the frame bundle $FM$ is always spin, ...
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eigenvalues of a first order differential operator on a manifold
I have the following, maybe naive question:
Given a smooth vector field $X$ on a smooth manifold $M$ with $\dim M \geq 1$.
Then this defines a linear operator
$$
D: C^\infty(M,\mathbb C) \to C^\infty(...
- 334
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Manifolds - Inverse Function Theorem Form?
For context, I am reading Appendix F is Milnor's book on Sullivan's No Wandering Theorem and this is where he uses the following manifolds result:
Let $F:X \rightarrow Y$ be a smooth function between ...
- 125
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Quotient of cohomology groups with different coefficents
Let $M$ be a smooth manifold, (if necessary I'm ok with assuming that $M$ is four dimensional, orientable, and closed). I wish to understand the quotient:
$$Q=H^1(M;\mathbb{R})/H^1(M;\mathbb{Z})$$
I ...
- 2,073
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Proving that the atlas of an open submanifold is smooth
$\def\AAA{\mathcal{A}}
\def\sbe{\subseteq}
\def\y{\psi}
\def\w{\omega}
\def\x{\chi}
$
In section $1.26$ of Lee's Intro to Smooth Manifolds the concept of an open submanifold is introduced, but it is ...
- 4,708
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Properly interpreting propositions 2.5 and 2.6 in Lee's Introduction to Smooth Manifolds
I want to make sure that I am correctly interpreting propositions $2.5$ and $2.6$ (picture below) of Lee's Introduction to Smooth Manifolds. Are the two following interpretations correct?
In $2.5$ (b)...
- 4,708
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Why is this subset associated to a $2$-tensor open and dense?
Let $S$ be a symmetric $(0, 2)$ tensor on a Riemannian manifold $M$. Define $E_S : M \to \mathbb{Z}$ by $E_S(x) = \left(\text{the number of distinct eigenvalues of } S_x\right)$. I've seen the ...
- 6,560
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Proving the image of a function of rank one is a curve using constant rank theorem
My question is based on this one and is prompted by my attempt to understand the constant rank theorem.
Specifically, suppose I have a smooth map $F : M \rightarrow R^k$ where $M \subset R^n$ is an $m$...
- 577
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39
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Why is $R_p:T_pM\times T_pM \to \operatorname{Hom}(E_p,E_p)?$
Let $M$ be a smooth manifold and $E\to M$ a vector bundle over $M$ and $\nabla:\Gamma(E) \to \Gamma(T^*M\otimes E)$ a connection on $E$.
The curvature $R$ of the connection is defined as $$R: \...
- 137
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Are there "differentiable manifolds" that don't admit a $C^1$-structure
It is well known that every $C^1$ manifold admits a smooth manifold structure. What if we relax the definition of smooth manifold so the transition maps need only be differentiable? Does every such &...
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