Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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Conformally immersed Riemann surfaces and foliations

I want to show that conformally immersed Riemann surfaces in $\mathbb{R}^4$ are leaves of a 2-foliation $\mathcal{F}$. I start with the generalized Weierstrass representation of the surfaces: take 4 ...
13
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2answers
1k views

Geometric interpretation of connection forms, torsion forms, curvature forms, etc

I have just begun learning about the connection 1-forms, torsion 2-forms, and curvature 2-forms in the context of Riemannian manifolds. However, I am finding it hard to relate these notions to any ...
3
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1answer
845 views

Invariant proof of the Contracted Bianchi Identity

In "Riemannian Manifolds: An Introduction to Curvature," John Lee states the following lemma: Lemma 7.7 (Contracted Bianchi Identity): The covariant derivatives of the Ricci and scalar curvatures ...
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1answer
240 views

Nonintegrable almost complex structures

The Newlander-Nirenberg theorem states that any Integrable Almost Complex manifold is a complex manifold. I am looking for natural examples of complex structures that are not integrable.
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2answers
155 views

Changing the manifold, preserving the discrete spectrum

On a Riemannian manifold $M$, the Laplace operator $L$ is uniquely defined. If $M$ is not compact, then $L$ admits a continuous spectrum. Is there a way of "changing" $M$ and/or $L$ in say $M'$ ...
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1answer
140 views

Finding a metric on a tubular neighborhood of an embedded surface

The setup for my question is an embedded surface $\Sigma\to M$ in a smooth, compact 4-manifold $M$. Assuming one knows the induced metric $g_\Sigma$ on $\Sigma$ , I would like to know if there is a ...
2
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1answer
331 views

Laplace-Beltrami Operator on Surfaces

I would like to know what is known about the spectrum of the Laplace-Beltrami operator on 2-dimensional negatively curved surfaces of constant curvature. For instance, What is the spectrum of the ...
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9answers
3k views

Introductory texts on manifolds

I was studying some hyperbolic geometry previously and realised that I needed to understand things in a more general setting in terms of a "manifold" which I don't yet know of. I was wondering if ...
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2answers
2k views

Which manifolds are parallelizable?

Recall that a manifold $M$ of dimension $n$ is parallelizable if there are $n$ vector fields that form a basis of the tangent space $T_x M$ at every point $x \in M$. This is equivalent to the tangent ...
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3answers
927 views

concrete examples of tangent bundles of smooth manifolds for standard spaces

I'm having trouble visualizing what the topology/atlas of a tangent bundle $TM$ looks like, for a smooth manifold $M$. I know that $$\dim(TM)=2\dim(M).$$ Do the tangent bundles of the following ...
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2answers
656 views

Draw a 3D parametric curve

I have many exercise about 3d parametric curve of class $C^{\infty}(\mathbb{R})$ of the form $$ \gamma(t) = \bigl( \gamma_x(t), \gamma_y(t), \gamma_z(t) \bigr)$$ (Example of curve: $\gamma(t) = ...
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2answers
532 views

Is $M=\{(x,|x|): x \in (-1, 1)\}$ not a differentiable manifold?

Let $M=\{(x,|x|): x \in (-1, 1)\}$. Then there is an atlas with only one coordinate chart $(M, (x, |x|) \mapsto x)$ for $M$. We don't need any coordinate transformation maps to worry about ...
4
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1answer
327 views

How does the boundary property usually work in PDE?

This may be related to the question Is the hypersurface of class $C^k$ a $C^k$-differentiable manifold?. In almost every chapter of the PDE textbook(e.g. Folland's Introduction to Partial Differential ...
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1answer
129 views

Is the hypersurface of class $C^k$ a $C^k$-differentiable manifold?

In Folland's Introduction to Partial Differential Equations: A subset $S$ of ${\mathbb R}^n$ is called a hypersurface of class $C^k$($1\leq k\leq\infty$) if for every $x_0\in S$ there is an open ...
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2answers
187 views

How to understand “maximal” in the definition of differentiable structure

Consider the definition of differentiable structure (Lectures on Differential Geometry, S.S. Chern): Suppose $M$ is an m-dimensional manifold. If a given set of coordinate charts ${\mathcal A} = ...
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2answers
177 views

Equivalent definitions of $C^r(\Omega)$?

The question is motivated from the definition of $C^r(\Omega)$ I learned from S.S.Chern's Lectures on Differential Geometry: Suppose $f$ is a real-valued function defined on an open set ...
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2answers
985 views

Can “being differentiable” imply “having continuous partial derivatives”?

Consider the following theorem: Let $E$ be a subset of ${\bf R}^n$, $f:E\to {\bf R}^m$ be a function, $F$ be a subset of $E$, and $x_0$ be an interior point of $F$. If all the partial derivatives ...
2
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0answers
240 views

fundamental group of closed surfaces as CW complexes

Let $T$ denote the $2$-torus, $P$ the projective plane, and $nT$/$nP$ the connected sum of $n$ tori/$n$ projective planes respectively. 1) how can I prove, that $nT$ and $nP$ are homeomorphic to ...
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1answer
74 views

Finding a path from one boundary component to the other

Let $X$ be a compact manifold with boundary $\partial X = X_0 \cup X_1$, and let $\omega$ be a volume form on $X$. Suppose $f:X \rightarrow [0,\infty)$ is a smooth non-negative function. Is it always ...
4
votes
1answer
512 views

approximating geodesic distances on the sphere by euclidean distances of a transformed sphere

Is there a way to find a function $F:\mathbb S^2 \rightarrow \mathbb R^3$ of class $C^1$, minimizing $$\int_{\mathbb S^2\times\mathbb S^2}(d(F(x),F(y))−\delta(x,y))^2 dx dy$$ , where $d$ stands for ...
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0answers
292 views

Question on Stokes' Theorem

Suppose $M$ is a smooth manifold and $f$ is a real valued smooth function on $M$. Set $N:=f^{-1}([0,1])$ and suppose $N$ is a compact submanifold of $M$. Let $\mu$ be a volume form on $M$ and $v$ a ...
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1answer
756 views

Piecewise smooth, non-orientable, closed-surface: a contradiction in terms, or am I going mad?

We had a lecture a few weeks back, looking at Gauss' divergence theorem, and in the definition of the theorem, it specified that the boundary of the volume under consideration, S, had to be a ...
4
votes
1answer
634 views

Smooth boundary condition implies exterior sphere condition

If $\Omega$ is a bounded domain in $\mathbf{R}^n$ with $C^2$ boundary, show that $\Omega$ satisfies "exterior sphere condition". Exterior sphere condition means that for each $z\in\partial\Omega$, ...
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1answer
414 views

exact differential n-forms

We know that a 1-form $\omega$ on a manifold $M$ is exact if and only if $\int_{\gamma}\omega=0$ for any closed loop $\gamma$. How can I prove the following generalization: $\omega$ is an exact n-form ...
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427 views

Gauss-Bonnet-Chern theorem

Good morning/day/evening/night, I was presented to the generalized Gauss-Bonnet-Chern theorem for hypersurfaces in Euclidean space; For a closed, even dimensional manifold $M$ with dimension $n$ ...
14
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1answer
454 views

Coordinate-free techniques in Lagrangian mechanics

Consider the following Lagrangian (Exercise 3.6B from Abraham and Marsden's Foundations of Mechanics): $$ L(\upsilon)=\frac12g(\upsilon,\upsilon)+V(\tau_Q\upsilon)+g(\upsilon,Y(\tau_Q\upsilon)) $$ ...
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0answers
127 views

homology Questions

I have some questions and would be infinitely grateful to you for your answers: 1- $f^{*}$ being the dual of $f_{*}$ so the degree (between top dimensional (co)homology groups) is the same for both ...
5
votes
2answers
635 views

What is the definition of tensor contraction?

According to Wikipedia's page on tensor contraction: In general, a tensor of type $(m,n)$ (with $m \geq 1$ and $n \geq 1$) is an element of the vector space $V \otimes \ldots \otimes V \otimes V^* ...
12
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1answer
2k views

precise official definition of a cell complex and CW-complex

I would be very grateful If someone could state a precise definition (direct one and inductive one) of a cell complex and CW-complex, since my intuition is telling that some restriction is missing and ...
11
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1answer
349 views

What is the universal property of the tangent bundle of a smooth manifold?

The process of writing my own notes on smooth manifolds have led me to wonder about this. All I've really found is the following: In addition to Madame Ehresmann's references, there is in ...
12
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1answer
681 views

Car movement - differential geometry interpretation

The problem presented below is from my differential geometry course. The initial reference is Nelson, Tensor Analysis 1967. The car is modelled as follows: Denote by $C(x,y)$ the center of the ...
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1answer
176 views

Normal coordinates vs. Locally flat

If $M$ is a Riemannian manifold the inverse function theorem tells us that for any $p \in M$ the exponential map gives us a nieghborhood $U$ of $p$ and normal coordinates $(x^i)$ in which the ...
8
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1answer
349 views

embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$

Consider the classic map $$F:\mathbb{RP}^2\rightarrow \mathbb{R}^4$$ defined by $$F[x,y,z]=(x^2-y^2,xy,xz,yz)$$. This defines a smooth embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$. It is clearly ...
4
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3answers
353 views

How to define a partial derivative invariantly?

Let $M$ be a smooth manifold and $f$, $g$ be smooth functions in some neibourhood of a point $x_0\in M$, $\nabla g\ne0$. 1) How to define $\displaystyle \frac{\partial f}{\partial g}$ invariantly? ...
4
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1answer
176 views

Trivializations of Tangent Bundle over 1-skeleton that do not extend to 2-skeleton, etc. Examples

I am reading up on Stiefel-Whitney classes $w_k(T_M)$ on $m$-manifolds; $k=1,2,\dots,m$, which are described as obstacles to extending a trivialization of a bundle, from the $k$-skeleton (assume ...
21
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1answer
472 views

functoriality of derivations

I seem to have problems understanding algebraically why given a map of manifolds $f: M \to N$ we get a bundle map $TM \to f^*TN$. Now, fiberwise it's all good. But I do not understand how to define ...
4
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1answer
153 views

Defining a Submanifold of $\mathbb{R}^n$

A submanifold (of $\mathbb{R}^n$), it appears, can be defined in several equivalent ways. One definition, paraphrased from Amann and Escher's Analysis II, is as follows: A subset $M$ of ...
3
votes
1answer
382 views

Global conformally flat coordinates in 2d spacetimes

Let $(M,g)$ be a 2 dimensional pseudo-Riemannian manifold that is topologically a disc. Is it possible to construct a global coordinate system in which the metric is conformally flat? I.e. coordinates ...
9
votes
1answer
812 views

How to apply Stokes' Theorem for manifolds with boundary

Original motivation: How can I apply Stokes' Theorem to the annulus $1 < r < 2$ in $\mathbb{R}^2$? Concerns: Since the annulus is a manifold without boundary, it would seem that Stokes' ...
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2answers
157 views

What does having a bar on a manifold mean?

If $M$ is a manifold then what is denoted as $\overline{M}$? I am guessing that it means a reversal of orientation. Related to the above I would like to understand the following construction of a ...
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4answers
325 views

Topologies and manifolds

This question might seem philosophical a bit: in a standard manifolds introductory course. when one talks about open , closed sets in $\mathbb{R}^n$ it's always the standard euclidian topology that ...
0
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1answer
187 views

injective immersions

if my function is an immersion and say it is defined on a path connected open of an euclidian space. immersions are locally injective. if we add the path connectedness could I assert that my ...
4
votes
1answer
408 views

local diffeomorphism

I hope this finds you all well. Could someone give me an example of a local diffeomorphism from $\mathbb{R}^p$ to $\mathbb{R}^p$ (function of class say $C^k$ with an invertible differential map in ...
15
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1answer
538 views

Group cohomology versus deRham cohomology with twisted coefficients

Let $G$ be a simple simply-connected Lie group, let $M$ be a 3-manifold and $P \to M$ a principal $G$-bundle. Let $A$ be a flat connection in this bundle, and let $\text{Ad} P$ be the associated ...
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2answers
143 views

“Orientation” of a surface boundary

Given is a surface patch embedded in $R^3$ with disk topology, i.e. it has a single boundary. I would like to know if it is possible to determine if the boundary is "interior" or "exterior" - an ...
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0answers
343 views

a question about the geodesic circle

How to show that the geodesic circles have constant geodesic curvature on a surface of constant curvature? Thanks
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0answers
381 views

A Way to make the following “proof” of the Hairy Ball Theorem rigorous?

I plan on giving a talk soon to undergraduates and I'd like to talk about the hairy ball theorem during the talk. I was trying to think of some sort of visually intuitive proof of this fact. (I ...
6
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1answer
148 views

What is the proper way to address this result?

Reading a paper I came through an argument proving the following: Let be given a smooth action of $\mathbb{R}^n$ on a manifold $M$, such that it is infinitesimally free and its orbits are ...
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2answers
152 views

rescaled metric quantities on rescaling metrics

I have the following basic, surely stupid, questions. Assume we have a Riemannian metric $g$ on a manifold $M$. let $a\in\mathbb{R}$ a constant and consider the metric $g_1=ag$. Which are the ...
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1answer
236 views

Prove that $TS^{n-1}$ is a trivial bundle, if $\mathbb{R}^n$ may be provided with an $\mathbb{R}$-algebra structure without zero divisors

Here I was shown how to prove that $TS^1$ is a trivial bundle. Similarly, I can show that $TS^3$ is a trivial bundle. Identify $\mathbb{H}$ with $\mathbb{R}^4$ and that that for $x \in S^3$ we have ...