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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Immersion, embedding and category theory

The map between smooth manifolds $f:M \to N$ is called an immersion if the tangent map $Tf$ is injective for each $x \in M$. There is a corresponding notion of submersion. About embeddings I met two ...
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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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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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1answer
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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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“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 ...
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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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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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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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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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What exactly is a Kähler Manifold?

Please scroll down to the bold subheaded section called Exact questions if you are too bored to read through the whole thing. I am a physics undergrad, trying to carry out some research work on ...
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48 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 ...
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Picture behind $SO(3)/SO(2)\simeq S^2$

Is there some kind of intuitive/waving hand argument to explain that $$SO(3)/SO(2) \simeq S^2 \; ?$$
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1answer
113 views

What is the equation describing a three dimensional, 14 point Star?

I need to model a 14 point star. This is a three dimensional surface where there is a point at each of the eight corners of a cube and each of the six sides. The object is uniform (i.e. planar ...
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1answer
58 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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1answer
37 views

Tensor Product of Vectors

let $S,T$ be respectively $k$-, $n$-tensors; $k,n>0$. Then we define the tensor product $$ T \otimes S(x_1,x_2,\ldots,x_{k+n}) := T(x_1,\ldots,x_k) S(x_{k+1}, \ldots, x_{k+n}) $$ (their product as ...
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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 ...
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101 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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1answer
59 views

A function on a space of symplectic forms

Let $M$ be a smooth manifold and $\text{symp}(M)$ be the set of all symplectic forms on $M$. Let $\text{Diff}_{ 0}(M)$ be a connected component of difeomorphisms of $M$. Then, is there an explicit ...
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113 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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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 ...
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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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22 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
37 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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172 views

Spivak Calculus on Manifolds, Theorem 5-2

In the proof Theorem 5-2 of Spivak Calculus on Mannifolds how is \begin{align*} V_2\cap M=\{f(a):(a,0)\in V_1\}? \end{align*} (That $\{f(a):(a,0)\in V_1\}=\{g(a,0):(a,0)\in V_1\}$ is clear.) Edit: ...
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1answer
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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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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 ...
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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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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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26 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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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 ...
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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 ...
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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 ...
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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
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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}
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+50

Jacobi field geodesic and calculus of variations.

How can we show that the second order variation to a geodesic is given by the Jacobi differential equation? In essence, \begin{equation} \frac{D^2}{dt^2}J(t)+R(J(t),\dot \gamma (t))\dot \gamma ...
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Classsifying 1- and 2- dimensional Algebras, up to Isomorphism

I am trying to find all 1- or 2- dimensional Lie Algebras "a" up to isomorphism. This is what I have so far: If a is 1-dimensional, then every vector (and therefore every tangent vector field) is of ...
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Non-compact complex manifolds which are not Stein

I am studying Stein manifolds, and it is clear for me that compact complex manifolds can not be Stein for obviously reasons. On the other hand, there exists some non-compact complex manifolds which ...
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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 ...
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Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
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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 ...
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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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34 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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What is the most general notion of “Fourier transform?”

I know the definition of a classical Fourier transform that maps a function f(x) on the real line X to a function F(p) on a dual space (here another real line and borrowing some physics notation) P. ...
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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 ...
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Stabilizer subgroup of 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 ...
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1answer
61 views

Show that $\omega_{ab}=-\omega_{ba}$ for a Riemannian connection

How can we see for the Riemannian connection, connection 1-form with its first index lowered $\omega_{ab}=\delta_{ac}{\omega^c}_b$ is antisymmetric in a, b, i.e. $\omega_{ab}=-\omega_{ba}$. Thanks.
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topic between algebra and geometry [on hold]

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 ...
3
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1answer
886 views

Pulling back vector fields

I want to find conditions under which one can pull-back vector fields (if it is at all possible). Let $F:M \to N$ be a smooth surjective map between two $C^{\infty}$ manifolds of the same dimension. ...
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1answer
45 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 ...
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1answer
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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 ...