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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derivative of a projection matrix

The projection onto a parametrised vector $v(\lambda)$ is $P_v = \frac{vv^{T}}{v^{T}v}.$ Its complement is $$P = I-\frac{vv^T}{v^{T}v}.$$ I've got an expression containing this complementary ...
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22 views

Hessian of the Stereographic projection

Consider the stereographic projection from the sphere $S^n$ onto $\mathbb{R}^n$, and take the usual local spherical (polar) coordinates $\omega_1,..,\omega_n$ on $S^n$ (coming from its embedding in ...
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39 views

Killing vector field for Witten complex?

I am reading a classical paper by Atiyah Bott "The moment map and equivariant cohomology". In paragraph "Relation with Witten complex" (at the very beginning of this paragraph) they claims that ...
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46 views

Hodge theory in general

I know a bit of Hodge theory, and I know that there is an analogue in the symplectic case, where instead of inducing the $\star$-product using the metric we use the symplectic form. Is in true in ...
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21 views

when can a surface conformally equivalent to the sphere be isometrically immersed?

Given a scalar function $s:S^2\to \mathbb{R}$, and the induced Euclidean metric $g$ on $S^2$, when can the sphere, equipped with metric $e^sg$, be isometrically immersed in $\mathbb{R}^3$? Is there a ...
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18 views

When does a non singular integrable differential one-form define a regular foliation?

Let $\mathcal{M}$ be a smooth manifold of dimension $m<+\infty$. Let $\theta$ be a nowhere vanishing (non-singular) differential one-form on $\mathcal{M}$ such that $\theta\wedge d\theta=0$. ...
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26 views

The first eigenvalue for Dirichlet boundary condition positive?

Let $M$ be a compact, n dimensional Riemannian manifold with boundary. Then we know that $W^{1,2}(M)=W^{1,2}_0(M)$, the latter is the completion of $C_0^{\infty}(M)$ function w.r.t $W^{1,2}(M)$-norm. ...
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16 views

A question about notation in derivatives inn $\partial_{\bar{A}}L^I$

In this paper http://arxiv.org/abs/1210.2332, when the authors say eq (2.17): $$\partial_{\mu}L^I=\partial_{\bar{A}}L^I\partial_{\mu}\bar{z}^A+\partial_AL\partial_{\mu}z^A$$, does they mean by ...
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24 views

Proving smooth map between smooth manifolds is constant based on push forward being zero

I have just me this problem in my class on smooth manifolds from Lee's introduction to smooth manifolds, from the chapter on the tangent bundle stating the following: Let M, N be smooth manifolds, ...
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45 views

normal bundles are stably isotopic

I am searching for a reference about the following theorem: Let $M \xrightarrow{i_{0}} \mathbb{R}^{k},\,\,$ $M \xrightarrow{i_{1}} \mathbb{R}^{l}$ be two embeddings of the manifold $M$. We know by ...
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21 views

tangent vectors

Let $S$ be any regular surface in $R^3$ and let $p \in S$ be any point. From Classical differential Geometry I Know that the tangent space of $S$ at p is a subspace of $R^3$. If I see the surface ...
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38 views

Checking a proof involving flows

I am going through the proof of theorem 2.12 of the book Lectures on the geometry of Poisson manifolds by I. Vaisman. It's just a bit of differential geometry, but as I don't use these methods very ...
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18 views

What is the most accessible reference on wall-crossing?

I am looking for a nice and easy to read reference on wall-crossing (in the context of Donaldson theory). Is there some accessible reference you have to suggest? I am interested in studying Donaldson ...
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12 views

Liegroup acting transitively => connected component acts locally transitive?

Suppose $G$ is a Liegroup acting transitively on a manifold $M$. Does that already imply, that the connected component $G^\circ$ of $G$ acts transitively on the connected components of $M$? I have a ...
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32 views

Is exponential map an immersion?

Let $M$ be a connected Riemannian manifold. For $p\in M$, the injectivity radius at $p$ is the sup of the $\epsilon >0$ such that the Riemannian-distance ball $B_\epsilon (p)$ is a geodesic ball, ...
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39 views

Relation between differential geometry and differential geodesy

I am not exactly clear on what are the differences between differential geometry and differential geodesy. Are principles in differential geometry used in differential geodesy ? It appears that ...
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45 views

Well definedness of Lie bracket (coordinates approach)

For practice I wanted to define Lie bracet in terms of coordinates and show that this definition is independet of the choice of coordinates. So I started with the definition ...
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36 views

What does it allow to see Differential Geometry from an abstract viewpoint?

I am currently learning Differential Geometry from the "categorical" point of view. We use sheaves, ringed spaces, group objects,... to define smooth manifolds, vector bundles,... My previous course ...
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17 views

Curves with distance between them growing locally as $o(d^k)$

Context: I'm searching for some standard definitions related to order of contact between curves (and smooth manifolds in general). My research has taken me to the concept of jets. Simply speaking, a ...
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47 views

Möbius strip parameterization and charts

A parameterization of the möbius-strip is given by : $$\begin{align}M=\{ (x,y,z) \in \mathbb R^3: x &= \cos t(1+ s\cos(t/2)),\\ y &= \sin t(1+ s\cos(t/2)),\\ z &= s\sin(t/2), \\ t ...
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39 views

Connection with zero torsion and curvature

I remember encountering the following theorem many, many time ago: if a connection $\nabla$ has $R = 0$ and $T = 0$ then... The problem is that I cannot remember the conclusion. Was it that the ...
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29 views

Is the curl of a vectorfield expressable in terms of a Lie derivative?

I am loving differential forms and the general version of Stokes theorem. However, I am also fond of the intuitive pictures of my older physics days of $grad$, $div$ and $curl$. I recently stumbled ...
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31 views

Diffusion on a Boundaryless Manifold and Tesselation

Suppose we are dealing with diffusion on a boundaryless manifold $M$. When people use the finite-element method to find an approximate solution, I always see them use \begin{align} \int_{M^*} W \Delta ...
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34 views

Is this subset of $\mathbb{R}^{3}$ a topological manifold?

Consider the set $\mathcal{M}_{1} = \{ \ (x,y,z) \in \mathbb{R}^{3}\ | \ y = -1 \ \}$, this is a plane. Also consider the set $\mathcal{M}_{2} = \{ \ (x,y,z) \in \mathbb{R}^{3}\ | \ x=y=0 \ \}$, which ...
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Unique function that gives an angle to a point of an interval.

(Sorry for my poor English in advance, as it is not my first language. Sorry for the vague title too, as I didn't know how to summarize the topic of the question). Here is an exercise that was ...
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26 views

“Scattering” in Riemannian spaces.

Let $(M,g)$ be a compact $n$-dimensional Riemannian space with boundary $\partial M$. Consider different types of "scattering" functions: 1) \begin{equation} y:TM \to \partial M: (x,v_x) \mapsto ...
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15 views

Weyl Transformations and Group actions

I have the following question. Let $(M,g_{ab})$ be a Riemannian manifold $M$ with metric $g$, and with an action of a Lie group $G$. Moreover, the Riemannian metric $g_{ab}$ is taken to be invariant ...
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28 views

Example for conjugate points with only one connecting geodesic

$\newcommand{\ga}{\gamma}$ $\newcommand{\al}{\alpha}$ I would like to find an example for a Riemannian manifold, that has two conjugate points $p,q$ with only one connecting geodesic between them. ...
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34 views

Manifolds, where its enough to have one chart for integration

Assume a compact connected manifold $M$ is given as a subset of some $\mathbb{R}^m$. Assume we have a chart $\gamma:U \rightarrow M$ such that $M-f(U)$ (the set $M$ without $f(U)$) has zero measure in ...
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46 views

How to show C is a regular curve?

The gradient of a differentiable function $f:S\to \mathbb R$ is a differentiable map grad $f:S\to \mathbb R^3$ which assigns to each point $p\in S$ a vector grad $f(p)\in T_p(S)\subset \mathbb R^3$ ...
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35 views

Proof of Synge theorem without homotopy

Can the following statement be proven without the use of homotopy? If $(M,g)$ is a closed, orientable Riemannian manifold with even dimension $m$ and positive sectional curvature then $M$ is simply ...
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32 views

Conformality of stereographic projection

I have to solve the following exercise: Let $S^m(r)=\{x\in \mathbb{R}^{m+1}|\sum_i (x^i)^2=r^2, r>0\}$ and let $\Phi$ be the Stereographic projection $$\Phi: ...
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14 views

Levi-Civita type Connection?

Let's assume $M$ is a real smooth manifold. Also, let $E$ is complexified Tangent Bundle $TM^c$ equipped with a hermitian metric structure. Then $E$ has complex Hermitian vector bundle structure. ...
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28 views

Problem from spivak's calculus singular cubes

For $r>0$ and $n$ an integer, define the singular $1$-cube $c_{r,n}:[0,1]\rightarrow \mathbb{R}^2-0$ by $c_{r,n}(t)=(r\cos(2n\pi t),r\sin(2n\pi t))$. Show that there exists a singular $2$ cube ...
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48 views

how to evaluate homotopy group of this specific structure

I am a Ph.D. student of physics and now I have some problems regarding the evaluation of homotopy group of a specific structure. In a paper, a specific topological structure is defined. The structure ...
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24 views

Tangent space on a regular submanifold

Let $f:M\rightarrow N$ be a submersion between two smooth manifolds, and let $P=f^{-1}(q)$ where $q\in N$. I want to show that $T_pP=\ker f_{*p}$ $\forall p\in P$. I need some tips on where to start ...
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24 views

A relation with geodesics on surfaces of revolution

Let $S$ be a surface of revolution, $$\varphi : (s,\theta ) \longmapsto (f(s)\cos \theta , f(s) \sin \theta , h(s)),$$ with $(f')^2+(h')^2 \neq 0$ and $f>0$. And let $\gamma : t \mapsto (\varphi ...
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17 views

Show that the intersection point of the normals converge to a point on the trace of the evolute

Let $\alpha(t): I \to R^2$ be a regular parametrized curve. Assume that $k(t) \neq 0$. The evolute is defined as the curve: $$\beta(t)=\alpha(t)+\frac{1}{k(t)}n(t)$$ Consider the normal lines of ...
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37 views

Show that if $\omega$ is a 1-form differential, then $\left\vert\int_{C}\omega\right\vert\leq ML$

Show that if $\omega$ is a 1-form differential define on $U\subset\mathbb{R}^{n}$, $c:[a,b]\to U$ is a differentiable curve and $\vert\omega(c(t))\vert\leq M$, for all $t\in [a,b]$, then ...
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34 views

Some details on the tangent space of a manifold

Let $x$ be a point of some smooth manifold $(M,\mathcal U)$ of dimension $n$. Let $I_x$ be the set of all $U\in \mathcal U$ containing $x$. Define the relation "$\sim_x$" on $I_x\times \mathbb R^n$ by ...
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19 views

Pullback of a Hamiltonian

I understand that a Hamiltonian vector field $H$ creates a Hamiltonian flow $\phi_t$. Now, in order to prove that the Hamiltonian is conserved one uses the following \begin{eqnarray*} ...
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25 views

Description of complex conjugate $\operatorname{Spin}^c$ structure without cocycles

The following uses exclusively cocycle descriptions for spin and spinc structures which I would like to avoid. See for example Nicolaescu "Notes on Seiberg-Witten invariants", pages 40-41 for their ...
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36 views

Christoffel symbols for the Poincaré ball model

The metric tensor $g_{ij}$ of the Poincaré ball model is $$ g_{ij} = \frac{\delta_{ij}}{(1 - x_k x^k)^2} $$ where $\delta_{ij}$ is the Kronecker delta and $x^k$ are the ambient Cartesian ...
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25 views

Conditions about Kahler metric

Suppose there is a Kahler metric $$ h=h_{i\bar{j}}dz^i \bar{dz^j}$$ then get $$ \frac{\partial h_{i\bar{j}} }{\partial z^k}=\frac{\partial h_{k\bar{j}} }{\partial z^i}$$ then claimed locally,there ...
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54 views

Structure of surface

We define surface patches $\sigma^x_{\pm} : U \rightarrow \mathbb{R}^3$ for $S^2$ by solving the equation $x^2+y^2+z^2=1$ for $x$ in terms of $y$ and $z$: $$\sigma^x_{\pm} (u, v)=(\pm \sqrt{1 - u^2 - ...
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28 views

Show that $dL_p(w)=L(w)$, can anyone please give some clue on how to proceed?

Prove that if $L:R^3\to R^3$ is a linear map and $S\subset R^3$ is a regular surface invariant under $L$, i.e., $L(S)\subset S$, then the restriction $L|S$ is a differentiable map and ...
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34 views

Polynomial functions on a smooth manifold

If one views $\Bbb R ^{2n}$ as the cotangent bundle of $\Bbb R ^n$, with coordinates $(q_1, \dots, q_n, p_1, \dots, p_n)$, then in order to do classical Hamiltonian mechanics on it one considers ...
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29 views

Coordinates on de Sitter space

I am trying to use a certain parametrization on de-Sitter space $dS^n$ and I am getting both the wrong scalar curvature and metric determinant. The formal definition of $dS^n$ in my work is ...
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31 views

first Chern class of pair (X,D)

Let $(X,D)$ be a pair of projective variety $X$ and $D$ is a simple normal crossing divisor on $X$ then is it correct that $$c_1(X,D)=c_1(X)+[D]$$ where $[D]$ is the current of integration
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30 views

Why do those terms vanish if the metric is Hermitian?

On this [page][1], the author says: We now define a Hermitian manifold as a complex manifold where there is a preferred class of coordinate systems in which unmixed components of metric tensor ...