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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Norm of clifford operator on forms

I am a beginner in differential geometry and would appreciate some pointers on how to answer the following question. Let $M$ be a closed orientable Riemannian manifold with $\{e^1,...,e^n\}$ an ...
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15 views

Representation of conjugate directions

Is there a way to represent conjugate directions on a Mohr circle of curvature? ( Surface Theory, Second fundamental form, M = 0 ) Directions given by double angles AOB, AOC. Is this attempt ...
3
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23 views

Holonomy computation in $S^2$

If $\gamma$ is a closed Loop in $S^2$ and $p\in S^2$, where $\gamma$ is the boundary curve of some region $X$ in $S^2$ (and $\gamma$ satisfied some regularity conditions), someone told me that the ...
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27 views

Spinors and forms

In this link http://benasque.org/2009gph/talks_contr/074Herdeiro.pdf page 15, it was said that: "Use spinorial geometry techniques: One takes the space of Dirac spinors to be the space of ...
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1answer
13 views

Countable intersection of Cut Locuses is always empty?

If $C_p(M)$ is the cut locus of some $p\in M$ in some Riemannian Manifold $M$, then does there always exist a countable collection of points $\{p_n\}_{m \in \mathbb{N}}$ such that: \begin{equation} ...
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1answer
54 views

Is Heisenberg group Euclidean?

I'm reading an article speaking about Heisenberg group $\mathbb H^n$ and some of its properties. Now, I have some questions to ask, hoping to be clear enought. Reading the introduction I've ...
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1answer
54 views

Are lightlike curves in the De Sitter space straight lines?

I think that every lightlike curve in $\mathbb{S}_1^2 \subseteq \mathbb{L}^3$ must be a line. But I'm having trouble concluding it. Let $\alpha\colon I \subseteq \Bbb R \to \Bbb S^2_1 \subseteq \Bbb ...
4
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21 views

How are the pseudo-Riemannian metric tensor properties restricted by the manifold topology in pseudo-Riemannian manifolds?

My understanding is that a pseudo-Riemannian metric tensor induces a topology that is not compatible with the manifold topology, and obviously the manifold topology prevails if we are to have a ...
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1answer
72 views

Easy classical physics made mathematically rigorous!

Consider the following: We are given a symplectic manifold $M$. Now, we define a Hamilton function $H : M \rightarrow \mathbb{R}.$ Additionally, we want that $H^{-1}(x)=:M_x$ is a submanifold. We can ...
3
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18 views

If $N$ is the boundary of Riemannian $M$, can I compute $i^{*}(* (\alpha \wedge \beta))$?

There wasn't enough room in the title to explain completely: $M$ is an oriented Riemannian manifold with boundary $N$. $\alpha$ and $\beta$ are differential forms on $M$, $*$ denotes the Hodge star, ...
4
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1answer
33 views

Classical differential operators with complex functions on Riemannian manifolds

I am having some trouble understanding how to use the classical operators ($\nabla, \operatorname{div}, \Delta$) with complex functions on a Riemannian manifold $(M, g)$. Consider the formula ...
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22 views

2-dimensional Riemann Manifold

I am looking for a proof of the theorem that states that any 2-dimensional Riemann Manifold is conformally flat in the case of a metric of signature 0, following through with Problem 6.30 in the text ...
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44 views

“Maximum point lies on a curve” implies tangential derivative is zero there.

Given a differentiable function $f:\mathbb{R}^2\to\mathbb{R}$, suppose that it has a local maximum at the point $(x_0,y_0)$. Let $\gamma$ be a smooth curve passing through $(x_0,y_0)$. Does it follow ...
2
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1answer
71 views

Doesn't $x^3+2y^3+3z^3=0$ give a surface in $R^3$?

In my last exam on Advanced Calculus (following Spivak's Calculus on Manifolds), I couldn't solve the following question. True or false: the set $S$ in $R^3$ given by $x^3+2y^3+3z^3=0$ is a ...
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1answer
23 views

Characterizing a surface

can somebody help me get started with this problem? I don't even know how to start the proof. Say $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable. Prove that $z=xf(y/x)$ belongs to a surface ...
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35 views

Total derivative involving rigid transformation

This stems from considering rigid body transformations, but is really a general question about total derivatives. Something is probably missing in my understanding here. A rigid body motion ...
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0answers
32 views

Computation in Wikipedia's article “Riemann Curvature Tensor”

This Wikipedia article explains how the Riemann curvature tensor is a measure of the failure for a tangent vector to parallel translate back to itself along an infinitesimally small loop. The article ...
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1answer
29 views

Understanding Symmetric tensor field

I am reading an article in which author calls some basic tensor analysis result. He states in general we define on $\mathbb R^N$ that $$ \mathcal T^k(\mathbb R^N):=\{\xi:\,\mathbb ...
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50 views

Cylinder and Möbius strip as fiber bundles: trivializations and cocycles

I know that this question has already been asked, but I couldn't find a clear answer. I have to show that the cylinder and the Möbius strip are fiber bundles over $S^1$ with fiber an open interval ...
3
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1answer
43 views

Tangent space of quotient space

Let $\pi : M \rightarrow M/G$ be the canonical projection, where $M$ is a manifold and $M/G$ is a quotient manifold. Now, what can we say about $d \pi (p) : T_pM \rightarrow T_p(M/G)$? From my ...
3
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1answer
64 views

Differential geometry in the context of manifolds

I am an undergraduate student of mathematics. I have a solid background on calculus, linear algebra, real analysis and point set topology, but I have never studied differential geometry. I am very ...
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2answers
142 views

Hodge Theory, intuition?

We have the following theorem of Hodge, as follows: $$\dim \ker \Delta^p = \dim H^p(M) = b_p(M).$$My question is, what is the intuition behind this statement?
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33 views

Differential geometry for nonlinear control theory

I am engineering student and I need to acquire a good understanding of some notions in differential geometry such as manifold, diffeomorphism, distributions etc.But I can't find a proper starting ...
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1answer
26 views

Lie group and stabilizer quotient

Let $G$ be a Lie group and $$G_x:=\{g \in G; Ad^*_g(x)=x\}$$ the stabilizer, where $Ad_g^*$ is the adjoint of the adjoint representation. Now, I was wondering why $G/G_x$ has a manifold structure. ...
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1answer
39 views

Does the Morse-Bott index of a critical point depend on the choice of metric?

By the Morse lemma there exists a coordinate chart $(x_1,...,x_n)$ in the neighbourhood of a critical point $p$ of a Morse function $f : M^n \to \mathbb{R}$ such that \begin{equation*} f(x) = f(p) - ...
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1answer
16 views

Flow of time-depended vector field

Suppose $X_t$ is a time-depended vector field with flow $\phi_t$, so, $\frac{d}{dt} \phi_t = X_t(\phi_t)$. Is it true that $d \phi_t(X_t(x)) = X_t(\phi_t(x))?$ This is true when $X_t$ does not ...
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16 views

Relation between torsion in torsion free of covariant derivative and torsion free group

Is there a relationship between "torsion free" of covariant derivatives and the torsion free group? Or is this just coincidence that people use the term "torsion free" here? It is in general required ...
2
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0answers
19 views

Accounting for signs in divergence thm. on Lorentzian manifold

I am trying to learn about integration in Lorentzian manifolds (I will use signature -+++) and have some problems. Oft quoted (in books for GR) form of divergence theorem is: $\int _U div( X ...
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27 views

Condition (C) of Palais-Smale

In Klingenberg's Notes, he makes the following definition: $\Lambda M$ will be said to satisfy the condition (C) of Palais-Smale if: Given a sequence $\{c_m\}$ on $\Lambda M$ satisfying: ...
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1answer
74 views
+50

Studying the family of curves $\beta(s,r) = \alpha(s) + r\,{\bf N}(s)$

I'm reviewing some stuff on plane curves, just because, and I would like to confirm some things. The whole exercise is: Let $\alpha(s) = (x(s),y(s))$ be a unit-speed parametrized curve, ${\bf ...
3
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1answer
39 views

Diffeomorphism between $\Bbb{R}^{4}$ and the cube

I'm looking for an explicit diffeomorphism between the four-dimensional euclidean space $\Bbb{R}^{4}$ and the four-dimensional open cube. I wonder whether there is a simple looking map, with simple ...
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1answer
24 views

Intersection of Cut Locuses

If $C_p(M)$ is the cut locus of some $p\in M$ in some Riemannian Manifold $M$, then when is: \begin{equation} \bigcap_{p\in M} C_p(M)=\emptyset\text{ ?} \end{equation}
2
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2answers
47 views

About gauge transformation

If $E$ is a vector bundle with a bundle metric, so we have ${\rm Aut}\ (E)$ whose fiber at $x\in M$ is the group of orthogonal transformation in $E_x$. Then gauge transformation is a section of ${\rm ...
8
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1answer
102 views

Riemannian metrics and how spaces look

I thought I had a fairly good understanding of Riemannian metrics until I came across this exercise in Petersen's book. Construct paper models of the Riemannian manifolds ($\mathbb{R}^2, dt^2 + ...
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0answers
36 views

length of continuously differentiable curves

I saw that the length of a continuously differentiable curve $\gamma$ in $\mathbb{R}^n$ with $\gamma(t) \neq 0$ is defined as $\int_a^b |\gamma^{'}(t)|dt$, as can be found here ...
6
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1answer
71 views

Flow of sum of non-commuting vector fields

Let $V,W\in\Gamma(M)$ be any two vector fields. Is there any "nice" expression for the flow of $V+W$ in terms of the flow of $V$ and the flow of $W$? It would be sufficient for me to have some sort of ...
1
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1answer
47 views

Orbits form a manifold?

A prominent example are the coadjoint orbits $O_x = \{Ad_u^*(x);u \in G\}$ where $x \in \mathfrak{g}$ and $G$ a Lie group with Adjoint map $Ad.$ Could anybody give me an easy argument why $O_x$ is a ...
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0answers
36 views

Exponential map only for matrix Lie algebras?

Recently, I stumbled over some proofs in Lie algebra theory and noticed that they often use the notion of an exponential map $e^{t \zeta}$ for $\zeta \in \mathfrak{g}$ such that $e^{t \zeta} \in G$ ...
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0answers
36 views

Is there a deRham (co)homology for vector-valued differential forms?

Is there a deRham (co)homology for vector-valued differential forms? The deRham (co)homology of differential forms has been well-discussed and well-founded, along with the fact that for the exterior ...
1
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1answer
24 views

Lie algebra of affine linear maps

Let $G$ be the Lie group of affine transformations, $$\{x \mapsto Ax+b,A \in GL(n), b \in \mathbb{R}^n\}.$$ We can represent these maps as matrices $$\begin{pmatrix} A & b \\ 0 & 1 ...
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1answer
48 views

topic between algebra and geometry [closed]

I have to do an exam on Differential Geometry and my teacher wants that I prepare a choosen topic, outside lectures program, that I will talk about at the oral part of the exam. I am interested in ...
2
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0answers
33 views

Tangent space of coadjoint orbit

Let $\xi \in T_xOx$ be a tangent vector at $x \in O_x :=\{\mathrm{ Ad} _{u}^*(x); u \in G\}$ for $x \in g^*.$ ($g$ is the Lie-Algebra) Then I read that this $\xi$ can be represented as the velocity ...
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0answers
37 views

How to tell complex structures apart

Complex structures are rigid, yet weirdly flexible. For example, the Riemannian mapping theorem says that every non-empty simply connected open subset of $\mathbb{C}$ that is not $\mathbb{C}$ is ...
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Lie algebra affine transformations [duplicate]

Let $G$ be the Lie group of affine transformations $$\{x \mapsto Ax+b,A \in GL(n), b \in \mathbb{R}^n\}.$$ Then we can represent these maps as matrices $\begin{pmatrix} A & b \\ 0 & 1 ...
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0answers
14 views

Adjoint and coadjoint orbits

I just read that for the Lie algebras $\mathfrak{gl}(N),\mathfrak{sl}(N),\mathfrak{so}(N),\mathfrak{sp}(2N)$ the adjoint and coadjoint orbits coincide. Now, the adjoint orbits are $O_{\xi} = ...
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1answer
46 views

Killing form - strange definition

I was just reading about Killing forms. In my opinion, the definition of these forms is quite strange. I mean why would one define $B(X,Y) = \mathrm{tr} (\mathrm{ad} (X) \circ \mathrm{ad} (Y))$? I ...
2
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1answer
29 views

Sanity check: smooth structure of tangent bundle

Let $M$ be a smooth $n$ manifold and let $TM$ denote its tangent bundle $$ TM = \bigsqcup_{x \in M} \{(x,T_x M)\}$$ I am trying to put a smooth structure (atlas) on $TM$ using the atlas on $M$. But ...
3
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1answer
27 views

Adjoint representation is Lie algebra homomorphism

Let $T_g:=L_g R_{g^{-1}}: G \rightarrow G$ be the standard automorphism of a Lie algebra, then $Ad_g:=DT_g(e): \mathfrak{g} \rightarrow \mathfrak{g} $is called the adjoint representation. Now, I want ...
2
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1answer
65 views

Would this be a homology theory?

Consider a manifold $M$, and denote by $\Delta _p M$ the set of all submanifolds of dimension $p$ (with or without boundary) of $M$. Define $G_pM$ to be the free abelian group generated by $\Delta_p ...
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11 views

Stabilizer subgroup in adjoint action

Given $b \in \mathfrak{su}(n)$, how can I find the stabilizer $\text{stab}(b)$ for the adjoint action of $SU(n)$ on $\mathfrak{su}(n)$ given by $Ad_U(b) = UbU^{\dagger}$ without using coordinates? The ...