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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Symplectic group action

Let $(M,\omega)$ be a symplectic manifold. We say that a group action $\phi: G \times M \rightarrow M$ is symplectic if each $\phi(g,.)$ is a symplectomorphism. Now, I am going through some lecture ...
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Curvature of $K$-invariant connection (principal bundles)

Here is a proposition from Kobayashi & Nomizu's Foundations of Differential Geometry. I don't understand how they obtain the final line of the proof. They write: \begin{align} ...
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Is it possible to put a Ricci-flat metric on the $n$-sphere for $ n>4$?

I'm looking for references which discuss the possibility of putting a Ricci-flat metric on the $n$-sphere for $n > 4$. Thank you for any kind of help.
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Smooth function, which separates between a closed and a open set.

Let $M$ be a smooth manifold, $O\subseteq M$ an open subset and $B\subseteq M$ a closed subset, such that $closure(O)\subseteq interior(B)$ I think there must exist a smooth function $f\colon ...
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The configuration space of a linkage

I am now reading the book 'geometry and billiards' by Tabachnikov. In this book, he has such a problem: Can anyone solve the case (c)?
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1 parameter subgroups and Lie groups

I was just reading some lectures notes (that are not online available unfortunatley) on Lie groups and found that sometimes the author just says if he wants to prove something for all Lie group ...
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27 views

Consistent and rigorous definitions of curve, arc and path

While reading many books, textbooks or Internet and actually I'm getting very much confused about the issues related to definitions of arcs, curves, paths, their parametrizations and curve images. Is ...
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94 views

Tangent space of the space of compatible complex structures

Let $(M,\omega)$ be a symplectic manifold, and let $\mathcal{J}(M,\omega)$ denote the space of complex structures on $M$ which are compatible with $\omega$. I have been told the following fact: We ...
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Exercise 3.2 of Do Carmo's “Riemannian Geometry”.

I am trying to do exercise 3.2 of Do Carmo's Riemannian Geometry. After constructing the natural metric in the tangent bundle of a Riemannian manifold, he defines "horizontal vector to the fiber". Its ...
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Help with exercise 2, chapter 2.5 (the first fundamental form), in Do Carmo (diff. geo.) [closed]

the exercise is: Let $x(u,v)=(\sin{\theta}\cos{\phi}, \sin{\theta}\sin{\phi}, \cos{\theta})$ be a parametrization of the unit sphere $S^2$. Let P be the plane $x=z\cot{\alpha}, 0<\alpha<\pi$, ...
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Computation of Hyperkahler Metric using kahler forms

I am trying to compute a hypekahler metric using its Kahler forms. We can expand the $\omega_{\alpha}$ as $\omega_{\alpha}={h_{\alpha}}_{ab} dx^a\wedge dx^b$ in which $x^a \in (u,\overline{u};p,q)$ ...
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36 views

Homeotopy of Shrinking Manifolds

Let $M$ be a $n$-dimensional open manifold in $\mathbb{R}^n$. Let $B^n_k$ be the closed $n$-dimensional ball of radius $k$. Let $$N_k = (M^c \oplus B^n_k)^c$$ where $X^c$ denotes the complement of ...
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Possible mistake in *Curves and Singularities*, 2nd ed., by Bruce & Giblin, p. 74

Page 74 in Bruce & Giblin's Curves and Singularities, 2nd ed. contains the following passage: Let $f: I \to \mathbb R$ be smooth and define $\phi: I \times \mathbb R^2 \to \mathbb R$ by $\phi ...
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27 views

slice charts of subbundles

Definition. Let $ \pi :E \to M $ be a smooth vector bundle of rank $r$. A subbundle of rank $k$ is a disjoint union $E'$ of $k$-subspaces, one for each fiber $E_p$ , such that $E'$ is an embedded ...
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Change of variable and diffeomorphic surfaces?

Suppose two curves $\gamma$ and $\gamma'$ are diffeomorphic. Is the arc-length measure $ds_\gamma$ absolutely continuous to $ds_\gamma'$ with a positive derivative? ($ds_\gamma=\phi\, ds_\gamma'$ for ...
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The Pullback Bundle is an Embedded Submanifold of its Parent Space

$\newcommand{\mc}{\mathcal} \newcommand{\set}[1]{\{#1\}} \DeclareMathOperator{\pr}{pr} \newcommand{\at}{\big|} \DeclareMathOperator{\GL}{GL}$ Let $\pi:E\to N$ be a smooth vector bundle over a smooth ...
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38 views

Slight confusion about Riemann curvature, in specific about $\nabla_{[X,Y]}$

In what follows I always use Einstein summation convention. The Riemann curvature is defined as $$ R(X,Y)Z = \nabla_{X}\nabla_{Y}Z - \nabla_{Y}\nabla_{X}Z - \nabla_{[X,Y]}Z $$ Now, I want to ...
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Hyperbolic geometry and the Triangle Inequality

In Is the shortest path in flat hyperbolic space straight relative to Euclidean space? I answered by refering to the Triangle Inequality (https://en.wikipedia.org/wiki/Triangle_inequality , Euclid's ...
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integral of product of curve with itself in three dimensional

I have the next problem: Let $\gamma:[0,1]\to R^3$ differentiable curve piecewise, and let $\Delta_r=\{(s,t)\in [0,1]^2| |\gamma(s)-\gamma(t)|<r\}$, i want to know if: ...
2
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1answer
33 views

Integration of $V$-valued differential form

When studying fibre bundles, connections and gauge theories it is usual to consider vector-valued differential forms, like the connection one-form, or it's pull back by a local trivialization known as ...
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resonance in pde

(I migrated the question from overflow b/c I think it belongs here instead.) About a year back I was in Professor Henry Mckean's office and he told me alot of interesting things. One thing he told me ...
5
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1answer
86 views

Theorema egregium violated in dimension $n \ge 4$?

Gauß showed that for surfaces in $\mathbb{R}^3$ the Gaussian curvature ( = sectional curvature) is invariant under local isometries. This is known as the thema egregium. Now in another question ...
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Does $\omega \wedge \mathrm{d} \omega=0$ (where $\omega$ is a non-vanishing $1$-form) imply $\mathrm{d} \omega \in \langle \omega\rangle$?

Let $\omega$ be a non-vanishing (for clarification: nowhere vanishing) smooth $1$-form on a smooth manifold $M$, if $\mathrm{d}\omega \wedge \omega =0$, do we already have $\mathrm{d}\omega= \sum a_i ...
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A Lie Group Homomorphism $f:G\to H$ Induces a Functor from Principal $G$-Bundles to $H$-Bundles

I am trying to understand Qiaochu Yuan's answer to this question. The first line of the answer reads: A Lie group homomorphism $f:G\to H$ induces a functor from the category of principal ...
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1answer
33 views

(Locally) sym., homogenous spaces and space forms

We had some definitions of particular types of Riemannian manifolds in our lecture 1.) Locally symmetric spaces. They were Riemannian manifolds with the property that $\nabla R=0$ everywhere. 2.) ...
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43 views

How To Formalize the Fact that $(g, h)\mapsto dL_g|_h$ is smooth where $g, h\in G$ a Lie Group

Let $G$ be a Lie group. I am wondering if there is a way to say that the map $(g, h)\mapsto dL_g|_h$ defined on $G\times G$ is a smooth map (Here $L_g$ is the left translation map from $G$ to $G$ ...
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1answer
63 views

Meaning of “locally homeomorphic to $\mathbb{R}^{n}$”

I am fairly new to differential geometry and approaching it with a physics background (in the study of general relativity), as a result I'm having a few struggles with terminology etc, so please bear ...
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47 views

Rigorously what is this integral?

I've been studying some gauge theories approach to problems in mechanics in order to get a better understanding of the ideas from gauge theories and to see some applications of fibre bundle theory. ...
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45 views

Lie Group Structure on the Spheres [duplicate]

Show that the spheres admitting Lie group structure are $S^0, S^1, S^3, S^7$ and give their Lie structures directly.
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19 views

Tilting sine function to get countably infinite nonregular values?

Let $f: \mathbb R \to \mathbb R$. A nonregular value $y$ of $f$ is any value such that not all $x \in f^{-1}(y)$ are regular. A point is regular if the Jacobian at it is surjective, in this case, has ...
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On geodesics in Schwarzschild spacetime

I am required to show that a circular lightlike geodesic exists in the Schwarzschild spacetime, and to find its radius. What's the best way to start this?
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Constructing coordinate maps on manifolds

I've been studying differential geometry for a little while now, but I've never properly justified to myself rigorously the need to consider other more general coordinate maps, other than Cartesian on ...
2
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1answer
59 views

Centroid of manifold

The centroid of a compact manifold is defined by the following equation: $c(Y_a)$ is the centroid of the parametrized manifold $Y_a$ is the point in $\Bbb R^n$ whose $i^{th}$ coordinate is given by ...
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2answers
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Divisors of differentials.

Let $\mathbb{C}$ be the base field. Suppose $C \subset \mathbb{P}_2$ is a nonsingular projective curve of degree $d > 3$. Must it be the case that $D \in \text{Div}(C)$ is the divisor of a ...
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Why “local holonomy contained in $SU(n)$” is equivalent to “vanishing Ricci curvature”?

I found it on Calabi-Yau manifolds' wiki page (https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold#Definitions) and can't figure out why is it true.
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1answer
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Differential Forms on Surfaces. Show that $N\cdot (\nabla \times V)\eta=d\phi$ on $x(D)$.

Let $M$ be an orientable surface in $\Bbb R^3$ with a unit normal vector field $N$ and let $x: D\to M$ be a patch. Let $\eta$ be a differential 2-form on $x(D)$ defined by $\eta(x_u,x_v)=\pm\|x_u ...
5
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1answer
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Is an injective morphism from a Lie group to itself surjective?

I have a question about Lie groups. Let $G$ be a finite dimensional (real or complex) Lie group and $f:G \rightarrow G$ an homomorphism of Lie groups. Edit : in the view of some counter-examples let's ...
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1answer
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Show by definition that $M=\{(x,y,z)|36x^2+4y^2-9z^2=36\}$ is a surface in $\Bbb R^3$

Show by definition that $M=\{(x,y,z)|36x^2+4y^2-9z^2=36\}$ is a surface in $\Bbb R^3$. Definition A surface in $\Bbb R^3$ is a subset $M$ of $R^3$ such that for each point $p$ of $M$ there exists a ...
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Why is a circle in a plane surrounded by 6 other circles?

When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other numbers? I'm ...
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2answers
74 views

Riemannian geometry vs Hyperbolic geometry

I am learning differential geometry in this semester. Concerning the riemannian geometry, if the cross-sectional curvature (riemannian metric ) is negative at every point, the manifold which arises is ...
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33 views

Second contracted Bianchi identity

Considering the Riemann tensor and the onece contracted second Bianchi identity $g^{ls}\nabla_sR_{ijkl}=-\nabla_iR_{jk}+\nabla_jR_{ik}$ why should it hold true that $g^{ls}\nabla_sR_{ijkl}=0$? In ...
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1answer
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Ricci flow on surfaces : step in proof

I am trying to realize the paper of richard hamilton's ricci flow on surfaces from the book of benett chow's Ricci flow : An Introduction.Here Hamilton denoted the trace free part of the Hessian of ...
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1answer
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Applications of Principal Bundle Construction: Vague Question

I recently read the principal $G$-bundle construction on a smooth manifold $M$, where $G$ is a Lie group. To understand them better, I am looking for some applications. Can the principal ...
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1answer
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Exterior derivarive dependent only on point

For any one-form (a linear form on the tangent space of each point) we have its exterior derivative $d\omega$ which is a two-form defined by $d\omega(X,Y)=D_X(\omega(Y))-D_Y(\omega(X))-\omega([X,Y])$ ...
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Non-English-language graduate-level textbooks on differential geometry

I'm looking for modern graduate-level non-English-language differential geometry textbooks. I'm interested in original works by non-English-language speakers in their native languages, not ...
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All possible flat conformal metrics of dimension greater than 2

Combining List of formulas in Riemannian geometry and Conformal symmetry, is there a proof which states $$ x^\mu \to \frac{x^\mu-a^\mu x^2}{1 - 2a\cdot x + a^2 x^2} $$ represents all possible ...
3
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1answer
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Is every open cover of a smooth manifold finer than a cover built from the union of disjoint open sets?

Let $M$ be a finite dimensional smooth manifold and $M=\bigcup_{i\in I}U_i$ an open cover of $M$. Does there exist a finite open cover $M=\bigcup_{k=0}^l V_k$, such that each $V_k$ is the disjoint ...
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Motivation for the name “vertical subspace” in the context of fiber bundles.

Let $p:E\to B$ be a smooth fiber bundle with fiber $F$. Consider the vector spaces $V_u=\{x\in T_uE: p_*(x)=0\}$. We call $V_u$ the vertical subspace of the tangent space $T_uE$. How can we see that ...
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Bounding distance between geodesics in manifolds with nonpositive curvature

I've recently read (in some notes by Mark Pollicott) the following related claims, which, although quite intuitive, I would like to see proven (and clarified). Let $M$ be a compact, connected ...
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Find the jacobian

I'm been struggling with the problem for a quite some time now. I need to find the jacobian for the following : $$u=x-y$$ $$v=xy$$ What I did : $$x=y+u\\x=\frac{v}{y}\\y=x-u\\y=\frac{v}{x}$$ ...