For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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4
votes
1answer
76 views

Orientability of submanifolds

What are some general conditions under which submanifolds of orientable manifolds will also be orientable. Of course this isn't true in general (for example, the Möbius band is a non-orientable ...
1
vote
2answers
112 views

lifting a product of commutators of standard generators on 2-manifolds

I have a problem with understand the proof http://www.ams.org/journals/proc/1972-032-01/S0002-9939-1972-0295352-2/S0002-9939-1972-0295352-2.pdf I don't understand this part: "(...) we can easily ...
0
votes
1answer
34 views

Simple closed curve definition of genus

The genus of a connected surface can be defined as the maximum number of disjoint simple closed curves that can be removed from it without disconnecting it. Why must the simple closed curves be ...
0
votes
0answers
22 views

Transversal Intersection

If $X, Z$ are manifolds of complentary dimension in $Y$, where $X$ compact, $Z$ is closed. Then show that in $ X \times Z, I_2(X \times \{0\},\{0\} \times Z) = 1$. Is this obvious because they ...
0
votes
0answers
15 views

The time evolution of levi-Civita connection

Assume a smooth one-parameter family of Riemannian metrics $g_{t}$. Write $h:=\frac {\partial}{\partial t}g$. In addition, assume that the Levi-civita connection on the Riemannian manifold ...
5
votes
3answers
256 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
64 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 ...
0
votes
1answer
23 views

Interacting Manifolds?

Usually, manifolds have certain internal properties which are being studied on their own. However, I was wondering if a field of mathematics exists where several separate manifolds are considered ...
4
votes
2answers
154 views

Why do folded concentric circles and rectangles form a hyperbolic paraboloid?

Here is a "self-forming" origami that I made from folding concentric circles - it would also happen if I folded concentric rectangles. How can the fold shapes such a saddle-like geometry?
1
vote
0answers
50 views

Seeking a good book or site describing the 3-sphere

would you be able to recommend a good book/chapter or a web site on visualization / structural elements / projection of the 3-sphere. I am trying to locate a good information source on this subject, ...
7
votes
3answers
140 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 ...
0
votes
1answer
24 views

Why are these meshes considered nonmanifold?

Why are these vertices and edges considered nonmanifold? What is the correct name for these surfaces(other than nonmanifold)?
2
votes
0answers
74 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 ...
3
votes
2answers
115 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
72 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
20 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
61 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
77 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, ...
1
vote
0answers
81 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 ...
2
votes
1answer
63 views

Givens rotation and retraction mapping

Here's part of the texts in the book Optimization Algorithms on Matrix Manifolds, when talking about the retraction on a matrix sub-manifold. A retraction is a mapping that maps a tangent vector in a ...
0
votes
0answers
28 views

Orientation of manifolds with boundary

I have an ambiguity about how to orient the boundary of a manifold. In particular : Consider the example $M=B^2 \subset \mathbb{R^2}$ be the manifold with boundary. suppose positive orientation for M ...
1
vote
1answer
70 views

Nonorientable surfaces: genus or demigenus?

The genus $g$ of a closed, orientable surface is the maximum number of disjoint simple closed curves that can be drawn on the surface without disconnecting it. In terms of the Euler characteristic, ...
0
votes
2answers
55 views

Lie bracket of vector fields on $R^2$

Compute the Lie bracket$$\Big[-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y},\frac{\partial}{\partial x}\Big]$$ on $R^2$ Can you help me please?
0
votes
1answer
30 views

Is there a smooth map from the square to the deltoid?

Is there a $C^\infty$ map between a unit square in $\mathbb R^2$ and a deltoid like this one The deltoid is obtained by varying the angles $\theta_1$, $\theta_2$ in the equations \begin{align} x_2 ...
2
votes
1answer
58 views

Question about vector fields and Lie group

Notation: $\chi(G)$ is the set of smooth vector fields on Lie group $G$, which in fact forms a vector space. Given a Lie group $G$, show that there exists a smooth vector field $X\in \chi(G)$, ...
1
vote
0answers
25 views

define distance in a manifold over the reals

G is a Hausdorff manifold over the reals with a finite atlas: $\exists m$ $G=\bigcup_{1 \leq i \leq n}U_i$, $g_i:U_i \to g_i(U_i) \subseteq \mathbb{R}^m$. Can I somehow define a metric inside G, ...
0
votes
1answer
25 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
37 views

Definition of tangent vector

I have a small bit of confusion with the definition my text is providing me with for a tangent vector. Given a manifold $M$, it is first stated that to define a tangent vector, a curve $c:(a,b) ...
5
votes
1answer
71 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
49 views

Is $(\mathbb{R},+)$ a smooth manifold?

I feel like $(\mathbb{R},+)$ is, but I'm not really sure. How would I know whether or not it is?
0
votes
1answer
46 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 ...
1
vote
0answers
50 views

Product Manifolds and Tangent spaces

Let $M\subset{E^{n}}$ be an r manifold and $N\subset{E^{m}}$ be an s manifold. Regarding $E^{m+n}$ as the Cartesian product $E^{n}\times{E^{m}}$, show that $M\times{N}$ is an (r+s)manifold. Show that ...
0
votes
1answer
38 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 ...
5
votes
0answers
80 views

Fixed Point Involutions

In recent reading on Riemann surfaces and complex manifolds (primary Miranda with a few random finds online), I encountered the notion of involutions, in particular fixed point involutions. We recall ...
2
votes
1answer
52 views

Proving a set of $2\times 3$ matrices is a manifold?

The way I have always been told to check if something is a manifold (I haven't had a whole lot of experience with them), is to check if the derivative of the function representing the loci of the ...
5
votes
1answer
115 views

Directional, differential and lie derivatives on manifolds intuition?

Trying to translate elementary multivariable calculus into the language of manifolds: Is the directional derivative on a manifold just a way of finding the rate of change of a vector in a single ...
1
vote
0answers
46 views

John Hempel's proof of residual finiteness of surface groups

John Hempel proved (http://www.ams.org/journals/proc/1972-032-01/S0002-9939-1972-0295352-2/S0002-9939-1972-0295352-2.pdf ) that fundamental groups of 2-manifolds are residually finite. I want to ...
3
votes
1answer
121 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 ...
-5
votes
1answer
787 views

Lee, Introduction to Smooth Manifolds Solutions

Does anybody know where I could find the solutions to the exercises from the book Lee, Introduction to Smooth Manifolds? I use the freely available online version ...
5
votes
1answer
90 views

Is a bounded (Riemannian) manifold always totally bounded?

Let $(M,g)$ be a finite dimensional Riemannian manifold. Suppose it is bounded as a metric space, i.e. $$\sup \{ d(x,y) | x,y \in M \} < \infty$$ where $d$ is the distance induced by $g$. Is $M$ ...
2
votes
2answers
117 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
72 views

Why is the vertex called non-manifold vertex?

I am working on triangle meshes in one 3D reconstruction project for a while. I know what one manifold vertex looks like and how to detect them. But I hope to understand the definition of non-manifold ...
0
votes
0answers
20 views

What is the connection between $\sqrt g$ and $|\det \psi'|$?

My text defined integration on a manifold as follows Let $M\subset \mathbb R^n$ be an $m$-dimensional manifold, $\varphi:U\to V$ a local map $(U\subset\mathbb R^m, V\subset M)$ and $f:M\to\mathbb ...
1
vote
0answers
52 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
51 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
61 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 ...
1
vote
1answer
37 views

General Linear Group is(not) compactly generated

We know that any connected Lie Group is compactly generated. I have a feeling that $\mathbb{GL}_n\mathbb{R}$ is not compactly generated. Is it true? If it is how can I prove this? If not, what ...
2
votes
1answer
59 views

Does the connected sum depend on direction of gluing?

The connected sum of two surfaces (2-manifolds) is defined by removing a disk from each and gluing the cut edges: (Image adapted from Wikipedia) Does the resultant surface (up to homeomorphism) ...
3
votes
1answer
191 views

Definition of Lie Groups

In the definition of Lie Group, we require that $$(x,y)\rightarrow x*y \text{ and } x\rightarrow x^{-1}$$ both be smooth. Are there any examples of groups that satisfy only one of these and not the ...
0
votes
0answers
32 views

Real analytic manifold

I found in some lecture notes such a definition of real analytic manifold: Let $X$ be a complex manifold (ringed space, locally isomorpic to...), $i: X \rightarrow X$ - a conjugation ...