4
votes
3answers
68 views

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. ...
0
votes
1answer
22 views

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 ...
6
votes
3answers
101 views

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 ...
1
vote
0answers
34 views

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 ...
1
vote
2answers
88 views

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? ...
2
votes
1answer
41 views

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 ...
2
votes
1answer
19 views

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 ...
2
votes
0answers
41 views

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 ...
1
vote
2answers
68 views

Proving a space is a manifold

Given a topological space defined as $A=A_1 \cup A_2$ with $A_1=\{(x,y) \in R^2 \space \space|\space \space x^2+y^2=1, x<0\}$, $A_2=\{(x,y) \in R^2 \space \space|\space \space |x|+|y|=1, ...
0
votes
0answers
45 views

How can we extend the proof that Lie groups are orientable to parallelizable manifolds?

Let $G$ be a Lie group with identity element $e$ and dimension $n$. I know we can take a non-vanishing $n$-form (call it $\omega$) on the vector space $T_eG$. For a Lie group, as we have an ...
0
votes
0answers
14 views

Conditions for the partition of unity in general topology

While I am reading "An Introduction to Manifolds" by Loring W. Tu, I come to see the above theorem. I followed the proof but got a question on (ii). We are talking about smooth manifolds. Why do we ...
4
votes
1answer
26 views

Slice of a coordinate system in a manifold

In the book - Foundations of differentiable manifolds and Lie groups by Frank Warner, the definition of a slice is as under. Suppose that $(U,\phi)$ is a coordinate system on $M$ (dimension $d$) with ...
0
votes
0answers
25 views

Finding a formula for a $C^{\infty}$ 1-form $\omega$.

Let me elaborate more. Suppose that $(U, x^1, ... , x^n)$ and $(V, y^1, ... , y^n)$ are two charts on $M$ with a nonempty overlap $U \cap V$. Then a $C^{\infty}$ 1-form $\omega$ on $U \cap V$ has two ...
0
votes
1answer
31 views

k+1 Differential form

Consider the k-form given by, $ w = \sum_{i_{1}<i_{2}<...<i_{k}} w_{i_{1}i_{2}...i_{k}} dx^{i_{1}}\wedge dx^{i_{2}}\wedge...\wedge dx^{i_{k}}$ Define $k+1$ form $dw$ , the differential of ...
3
votes
0answers
52 views

What is the kernel of a Maurer-Cartan form?

The Maurer-Cartan form on the Lie group $Gl(n,\mathbb{R})$ is a one-form taking values in $\mathfrak{gl}(n,\mathbb{R})$ as defined in the link. It has a rather concrete "extrinsic definition" as ...
2
votes
2answers
40 views

surface area of a Torus using differential forms,

Im studing integration over manifolds. I want to compute the surface area of the torus. Im given the usual parametrization $f(u,v)= ((a+b\cos(v))\cos(u), (a+b\cos(v))\sin(u), b\sin(v))$ for $0\leq ...
0
votes
0answers
27 views

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 ...
2
votes
0answers
31 views

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 ...
2
votes
1answer
30 views

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 ...
0
votes
1answer
34 views

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 ...
3
votes
1answer
27 views

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 ...
1
vote
0answers
24 views

Upper-half space of a manifold with boundary

Suppose that $X$ is a manifold with boundary and suppose that $x$ is a boundary point. Define the upper half space $H_x(X)$ in $T_x(X)$ to be the image of $\mathbb{H^k}$ under $d \phi_0:\mathbb{R^k} ...
2
votes
2answers
108 views

Expression for Laplace-Beltrami on sphere?

Is there a good expression for the Laplace-Beltrami on a function $u$ on a sphere or a circle of radius $R>0$ in terms of the Laplacian on ambient space? There is a formula on Wikipedia for the ...
0
votes
0answers
32 views

Example of 1-dimensional hypersurface in $\mathbb{R}^2$ which is compact?

Is there an explicit example of a $1$-dimensional $C^k$ hypersurface in $\mathbb{R}^2$ which has no boundary and is compact? I know of a circle, but want something like an interval.
0
votes
1answer
37 views

Smooth Embedding

I need help showing that the following smooth map is not a smooth embedding: $f:\mathbb{S}^1 \to \mathbb{R}$ defined by $f(z)= \operatorname{Re}(z)$. I know that this map is not a submersion because ...
3
votes
0answers
55 views

If there exists a diffeomorphism between two surfaces, what is the relation between Laplace-Beltrami operators on the surfaces?

Let $S(0)$ and $S(t)$ be a hypersurface in $\mathbb{R}^n$. Suppose there is a diffeomorphism $F^0_t:S(0) \to S(t)$. Suppose we have the Laplace-Beltrami operator $\Delta_{S(\cdot)}$. Let $u:S(t) \to ...
0
votes
1answer
14 views

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...
3
votes
1answer
36 views

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 ...
1
vote
0answers
21 views

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 \begin{equation} \Phi(X)=\sum_{j=1}^nV_jXV_j^* \end{equation} be a ...
4
votes
1answer
47 views

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 ...
0
votes
0answers
28 views

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 ...
2
votes
1answer
64 views

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 ...
1
vote
1answer
37 views

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 ...
2
votes
1answer
43 views

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 ...
3
votes
4answers
72 views

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. ...
0
votes
0answers
32 views

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 ...
3
votes
1answer
66 views

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 ...
2
votes
1answer
67 views

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 ...
1
vote
1answer
44 views

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 ...
-1
votes
1answer
19 views

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 ...
0
votes
0answers
49 views

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 ...
1
vote
1answer
50 views

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 ...
5
votes
0answers
74 views

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.
0
votes
0answers
38 views

Why does the expression of a point on a manifold seem to assume a coordinate system?

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 ...
0
votes
0answers
26 views

A question about complex polarization

Let $M$ be a smooth manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive dim$P\cap\bar P \cap ...
0
votes
1answer
36 views

Integration of forms over a smooth orientable manifold

Why does one needs to assume that a $n$-form has a compact support to be able to define its integration over a orientable $n$-manifold? I'm referring to the following text: ...
3
votes
1answer
54 views

Is this a geodesic?

Let $(M,g)$ be a riemannian manifold. Let $p$ in $M$ and $v,v_{0}$ two vectors in $\mathrm{T}_{p}M$. I am looking at the curve $$ \gamma \, : \, t \, \longmapsto \, \mathrm{Exp}_{p}(tv+v_{0}) $$ ...
0
votes
1answer
35 views

Compact Implies Closed

In general, i believe, it is not true that Compact Implies Closed. At least it is true that Compact does not imply closed and bounded. However, in case of differentiable manifolds, since they are ...
3
votes
1answer
72 views

Homework: calculation about differential form

Here is the question: Let $\omega = A dy\wedge dz + B dz \wedge dx + C dx \wedge dy$ in $\mathbf{R}^3$, and $d\omega = 0$. Denote \begin{eqnarray} \alpha = \int_0^1 tA(tx,ty,tz)dt\cdot(ydz-zdy)\\ ...
2
votes
2answers
59 views

Using a non-zero wedge product to write a set of vectors as a linear combination of another set of vectors in a finite dimensional space.

Question: Let $V$ be a finite dimensional vector space, and let $ \{ v_1, ..., v_r\}$ and $\{w_1, ..., w_r\}$ be two sets of vectors in $V$. Suppose that $\sum_{i=1}^{r} v_i \wedge w_i = 0$, and ...