Questions tagged [smooth-manifolds]

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

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26 views

Smooth vector bundles over contractible smooth manifolds with or without boundary

First, let me just write down the definitions I will use (see e.g. Lee - Introduction to Smooth Manifolds): Two (real) vector bundles $(E,B,\pi)$ and $(E',B,\pi')$ (over the same base space $B$) are ...
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18 views

Classification of smooth weak Fano three-folds

A variety is said to be 'weak Fano' or 'almost Fano' if its anti-canonical divisor is nef and big. In this question, let me restrict to the case of smooth varieties over the complex numbers. My ...
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29 views

Sectional curvature and Ricci curvature are bounded away from $0$ on compact manifolds of positive curvature

I've seen this claim a few times and it made sense in my mind but I realize I don't really know how to fully justify it: Let $M$ be a compact Riemannian manifold with sectional curvature (resp. Ricci ...
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13 views

Discreteness of branch points of a holomorphic map between Riemann surfaces [duplicate]

Let $X$ and $Y$ be Riemann surfaces (not necessarily compact) and $f:X\to Y$ a holomorphic map. A point $x\in X$ is said to be a ramificiation point of $f$ if the multiplicity of $f$ at $x$ is $\geq 2$...
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1answer
69 views

Are the elements of $C^\infty(M)$ smooth functions or equivalence classes of functions?

I am familiar with the notation $C^\infty_p(M)$, which denotes the algebra of germs of $C^\infty$ functions at $p$, where two functions defined on a neighborhood of $p$ are equivalent if they agree on ...
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1answer
47 views

Problem book recommendations on complex manifolds

I came across the book on Cauchy Riemann manifolds, "CR manifolds and tangential Cauchy Riemann complexes". The book does not have a problem section. I would be grateful if anyone recommends ...
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1answer
35 views

Simplifying the following computation of $n$-form

I'm trying to show $n-1$-form $$ \iota_{S^{n-1}}^*(\sum_{i=1}^n(-1)^{i-1}x^idx^1\wedge \cdots \widehat{dx^i}\wedge \cdots dx^n) $$ is a nowhere vanishing differential form on $S^{n-1}$. For the case $...
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1answer
39 views

Example of volume form

I know that a smooth $n$-manifold $M$ is orientable if and only if $M$ has a nowhere vanishing volume form. Since $S^n$ is a smooth orientable $n$-manifold, it has a nowhere vanshing volume form. Is ...
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58 views

Why is this a dense set? (Lemma 6.13, Lee ISM)

In Lemma 6.13 of Lee's 'Introduction to Smooth Manifolds' Lee is showing that the images of two functions $\kappa, \tau$ are subsets of measure zero in $\mathbb{RP}^{N-1}$. It follows then that their ...
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61 views

Coderivative of symmetric $2$-tensor

This is taken from a paper by M. Berger: Here $(X,g)$ is a Riemannian manifold and $h$ is a symmetric $2$-tensor on $X$. Could you help me to understand this definition, please? What does $\nabla^k$ ...
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32 views

A sub manifold of $\mathbb{R}^n$ is Riemannian?

Let $X$ be a smooth immersed submanifold of $\mathbb{R}^n$, with or without boundary. I have been told that then $X$ is a Riemannian manifold. I was explained that this follows from Nash embedding ...
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1answer
22 views

Showing straight lines through origin are geodesics of Poincare disc

Let $\mathbb D$ be the unit disc. The Poincare metric of $\mathbb D$ is given by $dx^2+dy^2/(1-x^2-y^2)^{-2}.$ I want to show that for $0\neq x\in\mathbb R\cap\mathbb D$ $\gamma_0(t)=tx$ is a geodesic....
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11 views

Methods of characterising $\mathbb{R}^{p,q}$ in terms of a general pseudo-Riemannian manifold?

The Question - Suppose we have a pseudo-Riemannian manifold $M$ with a pseudo-Riemannian metric of signature $(p,q)$. Then, what properties must $M$ additionally satisfy so that we get $M=\mathbb{R}^{...
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1answer
48 views

Smooth manifolds with a hole are homeomorphic

I'm trying to prove the following result: Let M be a smooth connected surface and $\psi_1,\psi_2:\mathbb{D}^2\to N$ two embeddings of the unitary 2-disk. Then, $M\setminus\psi_1(\mathbb{D}^2)$ and $M\...
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1answer
41 views

Is the image of a proper smooth embedding always a closed set?

Suppose $M$ is a $n$-dim. smooth manifold and $f : M \to \mathbb{R}^n$ is a smooth embedding of $M$ into some Euclidean space. I have two related questions. I have read on this site that a smooth ...
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1answer
15 views

Examples of zero-curvature Lorentzian manifolds apart from Minkowski space?

For an $n$-dimensional Lorentzian manifold $M$, it is a standard theorem in differential geometry that $M$ has a zero Riemann curvature tensor iff it is locally isometric to $n$-dimensional Minkowski ...
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1answer
27 views

Show a map from $\mathbb{P}^2$ to $\mathbb{P}^2$ is injective

I have a map $F: \mathbb{P}^2 \to \mathbb{P}^2$ defined by $F(x_0 : x_1 : x_2) = (x_0^2 + x_2^2 : x_0 x_1 + x_1 x_2 : x_0 x_2 + x_1^2)$ and I need to show it is well defined and smooth. I already ...
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1answer
39 views

$SO(3)$ double covers $L(4,1)$

Let $P^2$ be the real projective plane. I am trying to show that its unit tangent bundle (for a fixed arbitrary metric on $P^2$) is a lens space $L(4,1)$. It seems that this paper (https://www.maths....
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50 views

Definition of differentiable map that preserves orientations

I'm reading Frank W.Warner's "Foundations of Differentiable Manifolds and Lie Groups". In page 139, he defines the differentiable map that preserves orientations: Let M and N be orientable ...
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Constructing a smooth maximal atlas

Firstly consider the following notations and definitions: Notation: Let $(M,\mathfrak{A})$ be a smooth manifold. Then $\color{red}{\tau_M(\mathfrak{A})}$ denotes the topology in $M$ induced by the ...
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42 views

Berger notation for a triple product of tensors

The following definition is given in a paper by M. Berger: Here, $(M,g)$ is a Riemannian manifold and $S^2(M)$ is the space of symmetric $2$-tensors on $M$. Is there a coordinate-free equivalent for ...
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1answer
53 views

“Nice” metric on $\Bbb RP^3\#\Bbb RP^3$

Is there a standard riemannian metric on $\Bbb RP^3\#\Bbb RP^3$, in view of geometrization conjecture? It is known that $\Bbb RP^3\#\Bbb RP^3$ has $S^2\times S^1$ as a double cover, so its universal ...
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1answer
52 views

Derivative in the tangent space of a metric manifold

Consider a differentiable finite dimensional manifold $M$ with $\text{dim}M = n$, modelled on $\mathbb{R}^n$. Given a metric tensor $g$, at any point $p \in M$, the tangent space $T_pM$ can be ...
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18 views

How to understand local frame criterion for subbundles when a manifold has a boundary?

The following is from Lee's "Introduction to Smooth Manifolds" 2nd ed. Lemma 10.32 (Local Frame Criterion for Subbundles). Let $\pi:E\to M$ be a smooth vector bundle, and suppose that for ...
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48 views

Energy momentum tensor of a scalar field

Let $(M, g)$ be a Lorentzian manifold. A scalar field on M is a smooth function f : M → R. Its energy momentum tensor $Tf ∈ T_2^0(M)$ is $$T_f (X, Y ) = df(X) · df(Y ) − 12g(\operatorname{grad}f, \...
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60 views

Covariant derivative of tensor fields

Let $(M, g)$ be a semi-Riemannian manifold and $t \in T_s^r(M)$ be a tensor field whose coefficients in a chart. $(U, ϕ)$ are given by the functions $t^{i_1...i_r}_{j_1...j_s}:V \to R.$ Compute the ...
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36 views

Second fundamental form and the Hessian [closed]

Let f : Rn → R be a smooth function such that$ f(0) = 0 $and $Df|0 = 0. $Consider the Riemannian submanifold M := graphf = ${(x, f(x)) ∈ R^n+1 | x ∈ R^n }$ of $R^n+1$ equipped with the Euclidean ...
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1answer
51 views

Nondegenerate 2-forms exist on even-dimensional manifolds [duplicate]

Let $M$ be a smooth manifold. Recall that a 2-form $\omega$ on $M$ is called nondegenerate if for each $p\in M$ and $v\in T_pM-\{0\}$, there exists a $w\in T_pM$ with $\omega(v,w)\neq 0$. By linear ...
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31 views

doubt about smooth curves and pushforward

I'd like to clarify two things about pushforwards and smooth curves. $$\textbf{Question 1}$$ Suppose we have a smooth function $F:M\to N$, where $M$ and $N$ are two differentiable manifolds of ...
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1answer
36 views

Mapping degree mod $2$

Let $M$ be a closed smooth manifold, let $N$ be a conncected smooth manifold with boundary (perhaps $\partial N = \emptyset$), and let $f:M\to N$ be a smooth map. It is known that the number mod $2$ ...
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1answer
68 views

Derivative of a $1$-parameter family of Riemannian metrics

Let $S^n$ be a closed manifold and let $(M^{n+1},g)$ be a complete Riemannian manifold. Consider $\varphi: S \to M$ a fixed immersion and let $\varphi_t : S \to M$, $t\in(-\varepsilon, \varepsilon)$, ...
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1answer
35 views

Proof of collar neighborhood theorem

I'm trying to understand the proof of the collar neighborhood theorem given in the following document: http://www.math.toronto.edu/vtk/1300Fall2015/lecture-nov2.pdf At the end of the proof it says ...
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2answers
94 views

Compact manifold and eigenvalues

Let $X$ be a compact manifold and $f : X \to X$ a smooth map. Show that $\{x \in X | f(x) =x,\ 1 \text{ is not an eigenvalue of }df_x : T_xX \to T_xX\}$ is a finite set. If $1$ isn't an eigenvalue ...
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1answer
53 views

How to show that the tangent bundle is trivial?

So I was studying for a test and found this practice problem: Let $S^1=\{ (x,y)\in \mathbb{R}^2, x^2+y^2=1 \}$. i) Show that $S^1$ is a smooth manifold in $\mathbb{R}^2$ (Done) ii) Show that the ...
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10 views

Finding an equation for the closure of a parameterization

I'm working through Hubbard & Hubbard's Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. In the Chapter 3.1 (Manifolds) exercises, they ask to find an equation for the ...
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70 views

How to show exit time is continuous on unit sphere bundle $SM$?

Let $(M,g)$ be riemannian manifold. Unit sphere bundle $SM=\{(x,v):x\in M,|v|_g=1\}$. Given $(x, v) \in S M$, let $\gamma_{x, v}$ denote the unique geodesic determined by $(x, v)$ so that $\gamma_{x, ...
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35 views

Is a $1$-form $\omega :M\to T^*M$ smooth if, and only if, $\omega _p:T_pM\to \mathbb{R}$ is smooth and $\omega (p)=(p,\omega _p)$ for all $p\in M$?

Is a $1$-form $\omega :M\to T^*M$ smooth if, and only if, $\omega _p:T_pM\to \mathbb{R}$ is smooth and $\omega (p)=(p,\omega _p)$ for all $p\in M$? Below suppose that $M\subseteq\mathbb{R}^n$ is a ...
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41 views

Cotangent bundle and 1-form

Below are definitions that are associated with the tangent bundle. I would like to know the definitions analogous to those in the context of the cotangent bundle. I used the definitions below to study ...
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1answer
77 views

Why we define connection on a vector bundle in such an unnatural way? [closed]

From Wikipedia, we have: Let $E \to M$ be a smooth vector bundle over a differentiable manifold $M$. Let $\Gamma(E)$ be the space of all smooth sections. A connection on $E$ is an $\mathbb{R}$-linear ...
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1answer
38 views

Prove that flow map $\theta_t$ is orientation preserving

I was doing Lee's smooth manifold book exercise in Problem 15.4 needs to show Let $M$ be a oriented smooth manifold ,and $\theta$ the flow generated by some smooth vector field. Prove that $$\theta_t :...
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1answer
54 views

Doubt in the definition of continuous vector field

I'm wondering how the manifold topology on the tangent bundle $ TM $ yields a notion of continuity between tangent vectors as well, and not merely between base points. Let $X: M \rightarrow TM$ be a ...
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45 views

divergence operator on an oriented Riemannian manifold does not depend on the choice of oritetation

I was doing exerice in Lee's smooth manifold book,in page 423 needs to show that divergence operator on Riemannian manifold is independent of orientation ,and it's invariantly defined on all ...
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1answer
38 views

Why “compactness” assumption in the statement that $S$ is not boundary of some oriented compact smooth submanifold is necessary?

Let $M$ be a smooth manifold. Let $\omega\in \Omega^k(M)$ be the closed k-form on $M$.If we assume $\int_S\omega \ne 0$ on $S$(Assume $S $ is an compact oriented k-dimensional submanifold of $M$) ...
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1answer
24 views

What is the most succinct way to define vector fields and orientation of submanifolds in this context?

I am writing some notes on submanifolds of $\mathbb{R}^n$. I am reading several references in order to write short notes on this subject. The main definitions that I am using are below: Definition 1: ...
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1answer
32 views

Calculating with a specific Riemannian metric

I understand the idea of Riemannian metric, it takes to tangent vectors in a point $x$ of a manifold $M$ and defines an scalar product between them. My problem is how to compute the scalar product of ...
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16 views

Application of Yamabé theorem.

I have a question about an application of Yamabé's theorem which I recall: "Every arcwise connected subgroup of a Lie group is an analytic subgroup and therefore a Lie subgroup". My question ...
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1answer
44 views

Why $S^1$ is not boundary of compact orientable smooth submanifold in $\Bbb{R}^2-\{0\}$

Let $\omega = \frac{x}{x^2+y^2}dy - \frac{y}{x^2+y^2}dx$ is the smooth one-form defined on $\Bbb{R}^2\setminus \{0\}$, has nonzero integral over the $S^1$. Using stokes's theorem we can prove there ...
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2answers
124 views

Can any coordinate function be completed into an orthogonal coordinates system?

Let $U$ be an open neighbuorhood $U \subset \mathbb{R}^2$, and let $g$ be a smooth Riemannian metric on $U$. Let $x$ be a smooth function on $U$, with nonvanishing derivative $dx \neq 0$. Let $p \in U$...
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38 views

Understanding differential form of a manifold embedded in $\mathbb R^n$

In Example 16.9 of John Lee's Introduction to Smooth Manifolds 2nd edition, the following 2-form on $\mathbb{R}^3$ was considered: $$ w=x dy \wedge dz + y dz \wedge dx + z dx \wedge dy. $$ The ...
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1answer
59 views

Showing that a bundle homomorphism is a smooth isomorphism.

This is Lee's problem $10-11$ in his Introduction to Smooth Manifolds. If $\pi:E\to M$ and $\pi':E'\to M$ are vector bundles over the smooth manifold $M$ and if $F:E\to E'$ is a bijective smooth ...

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