# Tagged Questions

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### Prerequisite for Petersen's Riemannian Geometry

A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. ...
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### Definition of the integral of a vector field on Riemannian manifold and Euclidean spaces

Given a compact Riemannian manifold $(M,g)$ and a vector field $X \in \mathfrak{X}(M)$, is it possible to define the integral of $X$ on $M$? What if $M$ is a Euclidean space? Clearly the definition ...
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### let $\xi$ be an arbitrary vector bundle. Is $\xi\otimes\xi$ always orientable?

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that ...
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### Completeness of a Riemannian manifold with boundary

I have some issues understanding the notion of completeness of a Riemannian manifold with boundary. In the case of Riemannian manifolds without boundary, I found that completeness is usually defined ...
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### tensor product of two vector bundles

Let $\xi$ and $\eta$ be two vector bundles over the same base space $B$. Then $\xi\otimes \eta$ is orientable if and only if $\xi$ and $\eta$ are both orientable. How to prove this true or not true? ...
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### Finding relationship between Laplace-Beltrami operators of two spheres

Let $S$ and $T$ be spheres with radius $R_S$ and $R_T$ respectively. Define the diffeomorphism $\Phi:S \to T$ by $\Phi(s) = \frac{R_T}{R_S}s$. Given a function $u:T \to \mathbb{R}$, we can define ...
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### Confusion about $\Delta_{S^n} u(x) = \Delta u\bigg(\frac{x}{|x|}\bigg)$, why do we need to divide by $|x|$?

Let $S$ be a sphere of radius 1. We know the formula $$\Delta_S u(x) = \Delta u\bigg(\frac{x}{|x|}\bigg)$$ holds for a function $u:S \to \mathbb{R}$ where $\Delta_S$ is the Laplace-Beltrami and the ...
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### A few identities on differential forms

I just started studying differentiable manifolds, and I've encountered a few allegedly simple questions that aren't really simple to me. If possible, I'd be very happy to see full proofs (without ...
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### Sequence of critical values has no cluster point (Milnor, Morse Theory)

The following Claim is used in the proof of Theorem 3.5 in John Milnor's "Morse Theory": Claim: Let $f: M \rightarrow \mathbb{R}$ be a differentiable function on a manifold $M$ with no degenerate ...
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### Find the maximal integral curve $c(t)$ starting at the point $(a,b) \in \mathbb{R}^2$ of the given vector field.

Yet another integral curve problem. The vector field this time is $X_{(x,y)} = \dfrac{\partial}{\partial x} + x \dfrac{\partial}{\partial y}$. So, using what I learned from my last post, I should ...
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### Find the integral curves of the given vector field.

The vector field is as follows: $X_{(x,y)} = x \dfrac{\partial}{\partial x} - y \dfrac{\partial}{\partial y} = \begin{bmatrix} x \\y \end{bmatrix}$. I know that to find integral curves, you need to ...
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### Isometry of spheres/hypersurfaces and more generally Riemannian manifolds.

Let $M$ and $N$ be two spheres (of different radius) in $\mathbb{R}^n$ of dimension $n-1$. Suppose there is a Riemannian isometry between them (so a diffeomorphism and isometry). Then distances must ...
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### Second fundamental form without orientability?

Let $F$ be a $C^2$-hypersurface (or $n$-manifold) embedded in $\mathbb{R}^{n+1}$. Suppose $F$ is not orientable. Since I cannot choose a continuous global normal field, what consequences does this ...
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### If $S$ is a $C^k$ hypersurface, is $S\times (0,\infty)$ a $C^k$ hypersurface too?

Let $S$ be an $n$ dimensional $C^k$ hypersurface in $\mathbb{R}^{n+1}.$ Is $S \times (0,\infty)$ also a $C^k$ hypersurface (in $\mathbb{R}^{n+2}$)? I don't know what the chart map should be...
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### submanifold of Euclidean space is oriented if and only if normal bundle is an oriented vector bundle.

Let $f:M\longrightarrow \mathbb{R}^{n+k}$ be an immersion of $n$-dimensional manifold $M$ into $\mathbb{R}^n$. Let $\nu(M)$ be the normal bundle of $M$. Prove that $M$ is oriented if and only if ...
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### A question on the completely positive maps and manifold structure

I was reading a paper in which the curvature and Euler characteristic of a completely positive map (in finite dimensions). Let $$\Phi(X)=\sum_{j=1}^nV_jXV_j^*$$ be a ...
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### Passive and active coordinate transformation on a topological manifold.

Let us suppose we have $m$-dimensional smooth topological manifold $M$. Let $(U,\varphi)$ and $(V,\psi)$ be two charts on the manifold and $U \cap V \neq \emptyset$. For a point $p \in U \cap V$, we ...
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### Antipodal map and parallel transport on $S^3$

I'm reading Thurston and Levy's "Three-dimensional geometry and topology" where they have an informal explanation of what life in $S^3$ would look like. For the following detail that I am concerned ...
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### Tangent Space to a manifold

So, I have a manifold $M=\{\mathbf{x}:\mathbf{\Theta}\left(\mathbf{x}\right)=\mathbf{0}\}$. I can also write $M=\{\mathbf{x}:\mathbf{F}(\mathbf{x})=\mathbf{c}\}$. Both functions are differentiable. I ...
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### Normal vector of $\Gamma \times \mathbb{R}^+$ where $\Gamma$ is compact hypersurface

Let $\Gamma$ be a smooth boundaryless hypersurface of dimension $n-1$ in $\mathbb{R}^n$. Define $Q=\Gamma \times \mathbb{R}^+$. What does a normal vector of $Q$ look like? Because I want to compute ...
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### measure on non-oriented Riemannian manifold

Let $M$ be a non-oriented Riemannian manifold of dimension $m$. Nash embedding theorem implies that there exists an isometric embedding $\phi: M\longrightarrow \mathbb{R}^n$ for $n$ sufficiently ...
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### 1-manifold is orientable

I am trying to classify all compact 1-manifolds. I believe I can do it once I can show every 1-manifold is orientable. I have tried to show prove this a bunch of ways, but I can't get anywhere. ...
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### why does the differential of this function on the manifold have this expression?

I am studying the proof of a theorem from Warner, where I am stuck in the following place : Let $\psi:M\longrightarrow N$ be an immersion and, let $m\in M$. Let $(W,\tau)$ be a coordinate system ...
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### Recommendation on studying differential geometry

Below are what i studied so far: Rudin - Principles of Anlysis (only except one chapter, namely differential forms) Munkres - Topology (only point-set topology) Rudin - RCA (Only first 4 ...
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### Notation and hierarchy of cartesian spaces, euclidean spaces, riemannian spaces and manifolds

I am confused by some definitions. Forgive the looseness of my language. A Cartesian space is basically a space of points that can be represented by n-tuples ( and other things, but I won't go into ...
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### Bijective immersion is a diffeomorphism

Let $\psi:M\longrightarrow N$ be $C^\infty$, bijective immersion, the $\psi$ is a diffeomorphism. I am having trouble in proving this statement. What I have done so far is this : By inverse function ...
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### Is $f(x)+\sum_{p,i=1,…,m}\lambda_{p,i}x_{p,i}(x)$ globally defined?

Is the map $\Phi: \mathbb{R}^{N\cdot m}\times M\ni (\lambda_{p,i},x) \mapsto f(x)+\sum_{p}\sum_{i=1,...,m}\lambda_{p,i}x_{p,i}(x)\cdot \phi_{p}(x) \in \mathbb{R}$ globally defined, where M is ...
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### Manifold locally looks like a open set but not as a euclidean space?

I am reading about manifolds from the book by Millman. He says that manifolds locally looks like a open set, but there is no canonical way to make M look like a Euclidean Space, and so we can't define ...
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### Unique manifold structure

I am reading the first chapter from the book - Foundations of Differentiable manifolds and Lie groups by Warner. There, he has given two statements to be proved as exercises. a) Let $M$ be a ...
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### Does there exist a vector field s.t. all orbits are dense on $\mathbb{R}^2$

Is there a complete vector field such that the all orbits are dense on a contractible manifold? For example, $\mathbb{R}^2$,the interior of a $n$-unit ball.
This question came up when I was studying the definition of natural coordinate functions. In many books, such as O'Neil, natural coordinate functions are defined as $u_i(p) = p_i$ where ...