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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Is $z^2 = x^2 \cos y + 1$ an orientable surface?

So I can find one parametrization $\phi (u, v) = (u, v, \sqrt{u^2 \cos v + 1})$ which only does have of it. So now I need to find another parametrization which overlaps non trivally with $\phi.$ I ...
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43 views

Divergence of twice contravariant tensors

In a precedent topic (which can be found here : Relationship between divergence operators defined with respect to two different volume forms. ), I asked the question of the relationship between the ...
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63 views

$\mathbb{C}P^1$ diffeomorphic to $S^2$

I am trying to show that the complex projective line is diffeomorphic to the 2-sphere. I'm using the $C^{\infty}$ structure on $\mathbb{C}P^1$ given by $U_1 = \{ [z_1 : 1], z_1 \in \mathbb{C} \}$, ...
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75 views

On the Chern connection

It is well kown that If $\,\bar\partial_E$ defines a holomorphic structure on the complex vector bundle $E \to X$ and $K$ is a hermitian metric on (the fibres of) $E$, there is a unique connection ...
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52 views

Using the results of the local immersion/submersion theorems on manifolds

When $X,Y$ are $k$- and $l$-manifolds, we can have a function $f:X\rightarrow Y, x\in X$ such that $f$ is an immersion resp. submersion at $x$. The local immersion/submersion theorem now says: There ...
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29 views

A question about degrees of diffeomorphisms.

Let $\partial X$ be a compact boundaryless manifold and $Y$ be a connected manifold. It is known that if $f:\partial X\to Y$ is a diffeomorphism, and $f$ can be extended smoothly to $F:X\to Y$, then ...
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62 views

Definition of the first Chern class in terms of the Ricci form

From B, B & S - String Theory and M-Theory: What does the square bracket mean? Obviously since $\mathcal{R}$ is a form and $c_1$ is a number, $[.]$ has to be an operator on forms ...
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40 views

Lie Bracket, Hopf Fibration, independence of choice

Let $S^3$ be the standard sphere (with the metric induced from $\mathbb{R}^4$) and $\pi\colon S^3\rightarrow \mathbb{C}P^1$ the hopf fibration. Equip $\mathbb{C}P^1$ with the Fubini-Study metric so ...
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38 views

Any hint to show this flow property?

I just wonder if someone might help with this exercise. Let $X$ be a vector field on a compact smooth manifold $M$ and let $\phi_t$ be the flow of $X$. Show that for all $x \in M$, then ...
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44 views

Adjoint representation of the isotropie group of a homogeneous space

I have difficulties seeing why is the following true: Let $G$ be a lie group and $H$ a closed subgroup, with $\tilde{g}$ and $\tilde{h}$ their lie algebras. The adjoint action of $g\in G$ is given by ...
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45 views

Fundamental solution of Laplacian on manifold

I'm looking for a reference for the result that there exists a fundamental solution for the Laplacian on a flat torus $$\Delta \Gamma(x-y) = \delta(x-y), \quad x,y \in \mathbb T^2.$$ and that, ...
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130 views

How to Compute the Torsion and Curvature of a Parametric Curve

So I have a parametric curve $\bf{r}=${$x(n),y(n),z(n)$} such that the functions $x(n)$, $y(n)$ and $z(n)$ are polynomials of $4$-th degree. I have several of these curves, and I want to calculate the ...
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55 views

on special Kähler manifolds

Take Lie group $G$ with some hypotheses (compact, connected, semi-simple); call $T$ its maximal torus, its Lie algebra $\operatorname{Lie}(G)=\mathbf g$, its Cartan subalgebra ...
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53 views

The uniqueness of the Einstein metric on $\mathbb CP^n$

Is the Fubini-Study metric the unique Einstein metric (up to scaling by a constant) on $\mathbb CP^n$? More restrictively, Is the Fubini-Study metric the unique Kahler-Einstein metric (up to scaling ...
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17 views

Topology of the intersection of toric arrangement

Hope someone will help me in the solution of the following question. I'm working on some topological problem involving the topology of the intersection of some characters of the torus. I want ot find ...
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55 views

Are the principal curvatures on a surface always smooth?

It's easy to show that the principal curvatures on a surface are smooth away from umbilic points since we may write a expression for them using the Gauss curvature and the mean curvatures, locally. ...
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29 views

Ricci scalar algebra

Is this derivation correct? If $R=0$, $R$ being the scalar curvature, then: $$R_{;k}=0$$ $$(g^{ac}g^{bd}R_{abcd})_{;k}=0$$ $$g^{ac}g^{bd}(R_{abcd})_{;k}=0$$ $$R_{abcd;k}=0$$
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How do I determine sufficient conditions for the existence of the solution of an initial value problem?

Suppose that $f$ is a smooth function from $\mathbb R^{3}$ to $\mathbb R$ with $f(0,0,0)=0$. Under what sufficient condition will the differential equation $f(x,y,y')$ have a solution satisfying the ...
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94 views

measuring the curvature of a pringle

Assuming for the sake of argument that a pringle shaped potato chip has a constant negative curvature, (which according to http://math.stackexchange.com/a/617610/88985 it is not ) see also picture ...
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47 views

Principle Lines of curvature

In the text by Manfredo P. Do Carmo entitled Differential Geometry of Curves and Surfaces, an analysis of the principle directions is made near a non umbilic point on pp 160-161. I have followed his ...
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56 views

Why do we need an orientable surface for Gauss map?

I'm learning Differential Geometry recently with do Carmo's book. In the book, Gauss map is define as a differentiable map from an orientable surface $\mathcal{S}$ to $S^2$ in such a way that for ...
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29 views

$d+\Gamma$-understanding connections

$d+\Gamma$ is a formula commonly found in textbooks when one reads about connection. To be more precise: if $U$ is an open set over which a given vector bundle $E \to M$ is trivial then a coonection ...
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64 views

On Yau's (and Schoen's) proof of the positive mass theorem

I would like to face the proof of the positive mass theorem by Yau and Schoen. I have a Bsc in Mathematics and a Msc in Theoretical Physics and I'm preparing a PhD interview-challenge where I have to ...
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22 views

extension of a principal connection

I am trying to prove the following: Suppose that $\alpha:H\to G$ is a Lie group homomorphism and let $P\to M$ be a principal $H$-bundle and $Q\to M$ a principal $G$-bundle. Suppose further that there ...
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27 views

How do I take the inner product of these two tensors: $T^{ij}$ and $T_{ij}$

The tensors are of contravariant and covariant order two, respectively. Our teacher said something about the result being identity, or the kroneker delta $\delta_i^j$, I think, but I'm not too sure. ...
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28 views

parallel non-intersecting lines in E3

For time being I define a class of parallel lines in $ E^3 $ as lines with constant minimum distance along their common normal. Apart from helices with parametrization $ (x,y,z) = (a \cos (u) , a ...
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304 views

Computing the Fubini-Study metric

I would like to compute the Fubini-Study metric $g_{FS}$ for $\mathbb{CP}^{n}$ using as definition the metric induced from the round metric of the sphere by the Hopf fibration. I tried to compute on ...
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52 views

Naturality of Lie derivatives

It was left to me as an exercise to show the naturality of the Lie derivative of arbitrary tensors on a smooth manifold. Is the following argument correct (it seems too easy)? Let $X$ be a vector ...
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64 views

Norm equivalence of p-forms

Let $U$ be a bounded domain in the Euclid space $(\mathbb{R}^d,g)$. $g_{\wedge^p}$ denotes the fiber metric of $\wedge^pT^{\ast}\mathbb{R}^d$ derived from $g$. $A^p$ denotes the set of p- forms on ...
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107 views

Quaternion Calculus

I was reading a note on Quaternion(Link) and I am happened to read a section regarding a solution of quaternion differential equation. I am putting that segment as picture format here for more ...
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49 views

about moments of a uniform distribution on a high-dimensional ball

I need to understand how the following integrals depend on the dimension $d$; the result should be about a (negative) power of $d$. Let $\mathbb{B}^d$ be the $d$-dimensional ball of radius $1$, ...
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26 views

How do I map Df(w) to it's [lie] group/algebra representation?

E.G. For $p,w\in(\mathbb{R}^3,+,\times_\vartheta)$ with $(\mathbb{R}^3,+)$ a vector space and with $p=(r,s,t)$, $w=(x,y,z)$ where we have $p\times_\vartheta ...
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33 views

Sign of Laplacian in a N-Dimensional Space Under Other Specific Conditions

Consider that $f: \mathbb{R}^N \to \mathbb{R} $ is infinite times differentiable and is smooth over the entire domain. Take $N$ to be large. Also at point $\mathbf{x_0}$: $$ ...
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130 views

Is Hodge star operation can be understood as contraction after tensor product of a $p$-form with the volume element?

By defintion, the Hodge star of a $p$-form $\omega_{a_1\cdots a_p}$ on a $n$-dimensional manifold is given by $*\omega_{b_1\cdots b_{n-p}}=\frac{1}{p!}\omega^{a_1\cdots a_p}\epsilon_{a_1\cdots ...
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63 views

geometrical consequences of nonpositive or negative Ricci curvature.

well,that's pretty much the question. I'd like to know if somebody could point me out if there's any geometrical implications following an upper bound on the Ricci curvature on a riemannian manifold. ...
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56 views

Manifold characteristics in terms of Riemannian metric

I wonder what characteristics of Riemannian manifold can be expressed in terms of metric? Are there any results similar to Gauss–Bonnet theorem? Does the Riemannian metric give any information about ...
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52 views

Can the closure of a geodesic chart of a manifold without boundary (and with bounded geometry) be defined as a compact manifold with boundary?

Let $(M,g)$ be a $d$-dimensional smooth, riemannian manifold without boundary, which is geodesically complete, connected, has positive injectivity radius and a bounded geometry (i.e. all derivatives ...
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39 views

Invariant characterization of vector bundles associated to a principal bundle?

I have two related questions. Suppose I have a principal $G$-bundle $P\xrightarrow{\pi} M$. The usual construction of an associated vector bundle goes as follows. Fix some representation $\rho : ...
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50 views

differential of $f:X\to\Sigma$ as an elliptic surface,

Let $X$ be an algebraic surface surface and $\sum$ an algebraic curve, and assume, $f:X\to\Sigma$ be an elliptic surface, my question is Why the differential $df$ can be viewed as an injection of ...
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45 views

Definition - Logarithmic Zeros and Algebraic Vector Fields

In Algebraic Geometry, what is the difference between an algebraic vector field and a derivation (are they completely different or synonymous)? What does it mean to say an algebraic vector field has ...
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45 views

Reference Request for differential ideals of Pfaffian forms on jet bundles

My setting is the following: Given two families of differential forms $\omega^i$ (with $i=1,...,m$) and $\mathrm d F^j$ (with $j=1...n-m$), define $$\eta^i := \sum_{j=1}^m a^i_j\omega^j + ...
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50 views

Curve and Constant Curvature

I have initial position vector $p_0$, given curve-linear length $1$. It can be parameterized by $s\in[0,1]$. Assume we have the equation to generate the curve from given starting point and constant ...
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55 views

differential form: a question in Novikov's book

I met a problem when reading the book of Novikov et al. "Modern geometry, vol 1". In page 200 of the book (GTM 93, English version), they say for 2-form $F$, $$ (F\wedge F)_{0123}=-{1\over ...
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Which part of differential geomety uses metrization theorems?

I learned three metrization theorems last year, which are Nagata-Smirnov,Smirnov and Bing. I thought these theorems are purely topological theorems, but i recently saw a post which says these ...
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Mean curvature of even order

I read Antonio Ros, Compact Hypersurfaces with Constant Higher Order Mean Curvatures,1987. I don't understand following sentence from the second page 6th line. From the Gauss equation, we have ...
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Equation of a curve with a local minimum fixed at $x=a$ when we rotate the curve about the origin.

We have a strangely curved plank. If we place a round weighted object on it, it will rest itself at one point of it, when we incline the plank slowly, the object will gradually move towards a ...
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79 views

curvature of a plane curve

I'm trying to prove the formula to calculate the curvature of a plane curve. But I end up with the wrong sign and can't figure out why: I want to proof $\kappa(t) = \frac{\dot c(t) \cdot \ddot ...
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58 views

Gauge covariant derivative on principal bundle over $\mathbb R^d$

I try to understand the physical definition of covariant derivative in gauge theories in terms of the exterior covariant derivative of vector-valued forms defined as the horizontal projection wrt a ...
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106 views

Integrability of 1-forms and Stokes' Theorem

Let $\alpha$ be a $1$-form defined on a manifold $M$ and $\Delta = ker (\alpha)$. The classical theorem of Frobenius says that $\Delta$ is integrable if $\alpha \wedge d\alpha =0$ i.e if $d\alpha$ is ...
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A question about the boundary points of manifolds.

A topological $n$-manifold is a second countable Hausdorff space such that every point has a neighbourhood which is homeomorphic to an open ball centred at the origin in $\Bbb{R}^n$. The "centred at ...