Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

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confusion on notation and action

I was reading one lecture note, what is what is $v(x_{i}^{1})$ here how it is defined. Thank you. I couldn't find in before of the lecture note. Proposition A The tangent bundle TM of any given ...
Document123's user avatar
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When shall a curve that connects two points on a surface be a line?

I am preparing to teach 'cross-section of solid structures'. Currently I am studying the cross section of a cylinder. I take one dot $A$ on the top circle, and dot $B$ on the bottom circle. I think ...
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Which differential equations are invariant under change of camera projection?

For background, I am working in the plane $\mathbb{R}^2$. I know that the derivatives $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ are invariant under translation. I know that the ...
user326210's user avatar
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Scalar functions under diffeomorphism of domain

Consider the space of functions $F(\mathbb{R^n}) := \{f \in C^{\infty}(\mathbb{R^n})| \lim_{|x|\to \infty} f(x)=0 \}$. Consider the space of coordinate transformations on the domain $\mathbb{R^n}$, ...
Joeseph123's user avatar
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Hodge dual of dy in Minkowski space

I'm following the formula for the Hodge dual as follows $$ (\star \omega)_{\mu_1\cdots\mu_{m-p}} =\frac{1}{p!} \sqrt{|g|} \varepsilon_{\mu_1\cdots\mu_{m-p}\nu_1\cdots\nu_p} \omega^{\nu_1\cdots\nu_p} $$...
Liu Zhiyu's user avatar
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Pullback of vector field under inclusion map

Suppose we have a manifold $X$ embedded in ambient space (this is $\mathbb{P}^n$), $$ \iota: X \hookrightarrow \mathbb{P}^n. $$ Given a vector field on $\mathbb{P}^n$, is there any way, canonical or ...
Eweler's user avatar
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What is the pullback in the category of differential manifolds?

I have learned that the pullback exists in the category of topology spaces. I am wondering if the pullback exists in the category of differential manifolds. In fact I want to convince myself the ...
Zoudelong's user avatar
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About a surface connected

Let $S$ be a connected surface. In many propositions, exercises, etc., I have seen that if the surface is connected, given any point $p \in S$, then there exists a parametrization of the surface from ...
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Lee's definition of a consistently oriented basis for a real vector space $V$ of dimension $n\geq 1$

Let $V$ be a real vector space of dimension $n\geq 1$. Lee defines in his book Introduction to Smooth Manifolds chapter 15 that We say that two ordered bases $(E_1,\dots,E_n), (\tilde{E}_1,\dots,\...
Cartesian Bear's user avatar
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Compute the differential of a group action

Consider the map $$\phi: \mathbb{R}^*\to \text{Aut}(\mathbb{P}^2)$$ defined as $$ t\mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & t^2 \end{pmatrix},$$ where by $\...
ark's user avatar
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In which points is $f: \mathbb{R}P^2 \rightarrow \mathbb{R}^3$ an inmersion.

In this problem $\mathbb{R}P^2$ denotes the real proyective space of dimension $2$ which is obtained by identifying one dimensional spaces in $\mathbb{R}^3$. We denote by $\pi$ the usual projection ...
H4z3's user avatar
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A differential of a flow and the Jacobian of a vector field on a sphere

I have a vector field on $\mathbb{S}^{d-1}$ of the form $V_x = P_x(f(x))$ where $P_x(y) = y-\langle x,y\rangle x$ is the projection of $y\in\mathbb{R}^d$ onto $T_x\mathbb{S}^{d-1}$ and $f:\mathbb{R}^d\...
Kaira's user avatar
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Some questions about integration and operator theory.

As we all know there are multiple integral operators which all basically do the same thing in various contexts. I am talking about operators like the Lebegues integral, Riemann integral and more ...
Logarithmnepnep's user avatar
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Is geodesic convexity preserved under strereographic projection? [closed]

Let $\Omega$ be a geodesically convex subset of the hemisphere. Is it true that $\Pi(\Omega)$ is a convex set of $\Pi(S^2_{+})=B_1(0)$ where $\Pi$ denotes the stereographic projection from the south ...
Student's user avatar
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Given $\gamma=\gamma(t,s):I\times I\to M$ the vector fields $\frac{\partial \gamma}{\partial s},\frac{\partial \gamma}{\partial t}$ commute

The problem is pretty much stated in the title ($I$ is the interval $[0,1]$ and $M$ is a differentiable manifold). Clearly we can't use the fact that $\gamma_*[X,Y]=[\gamma_*X,\gamma_*Y]$, because $\...
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2 votes
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Defining the weak Einstein condition

Let $(E,h)$ be a Hermitian vector bundle over a compact Kähler manifold $(X,g)$. I'm trying to understand how Kobayashi defines the weak Einstein condition, but I'm not fully figuring this out. He ...
Rene's user avatar
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Proof of Cartan's Lemma in Lee's Smooth Manifolds

I am trying to prove problem 14-4 from Lee's Smooth manifolds. CARTAN'S LEMMA: Let $M$ be a smooth $n$-manifold with or without boundary, and let $\left(\omega^1, \ldots, \omega^k\right)$ be an ...
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Ehresmann connections on general G-bundles?

There are two definitions of connections on bundles in terms of horizontal bundles: In the case of a principal $G$-bundle $P$, a connection is a subbundle $HP<TP$ such that $HP\oplus VP=TP$ and it ...
Alex Bogatskiy's user avatar
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Are there isospectrally equivalent exotic spheres?

Let $X$ and $Y$ be two different exotic spheres. Are there metrics $g$ and $h$ on $X$ and $Y$, respectively, such that the laplacians of $(X,g)$ and $(Y,h)$ have the same spectrum?
discretephenom's user avatar
5 votes
2 answers
320 views

Flat Bundle vs Trivial bundle

In R.W. Sharpe's Differential Geometry, a flat fibre bundle is defined as a bundle whose transition functions are constant. I don't understand the difference between this and a trivial bundle. Because,...
Jarah Fluxman's user avatar
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Hint for parametrizing a surface that is equidistant from a plane through the origin and a point not on it

$\newcommand{\norm}[1]{\|#1\|}$ This is for homework, so please only hints and light nudges! Problem: Let $\mathcal{P}$ be the plane through the origin with normal vector $\vec{n}=(a,b, c)$ and if $F =...
Nap D. Lover's user avatar
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Reflection of the "Figure Eight" not diffeomorphic to the "Figure Eight"

In problem 3 of page 73 from "An Introduction to Differentiable Manifolds and Riemannian Geometry" by Boothby, he defines the "figure eight" as: $G(t) = \left( \left( 2\cos(g(t) - \...
Martin's user avatar
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Maximal differential structure on a finite dimensional vector space

Let $M$ be a finite dimensional vector space of dimension $N$. For a basis $\beta = \{v_1, ..., v_N\}$ of $M$, define the map $X_\beta : \mathbb{R}^N \to M$ by $X_\beta(a_1, ..., a_N) = a_1 v_1 + ... +...
Lucas Linhares's user avatar
3 votes
1 answer
169 views

A simple question about the Hodge star

The usual definition of the Hodge star says that it maps $\Lambda^k(V)$ to $\Lambda^{n-k}(V)$ in such a way that for each pair $\omega, \eta \in \Lambda^k(V)$ holds $\omega \wedge *\eta = \langle \...
tsnao's user avatar
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Why is the Jacobi equation not tensorial?

Let $\gamma: (-\epsilon,+\epsilon)\times I\to (M,g)$, $\gamma_s(t):=\gamma(s,t)$, be a family of curves in a Riemannian manifold $(M,g)$. Denote $J:=\partial_s\gamma\rvert_{s=0}$ and $\gamma_s':=\...
Haydn86's user avatar
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Why are Klein geometries flat?

A Klein geometry can be seen as a principal $H$-bundle of the form $G\to G/H$. It is known (cf. Sharpe and Chern's book) that Klein geometries are examples of flat Cartan geometries because their ...
Alex Bogatskiy's user avatar
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Integral formula for Hopf map in local coordinates appears to vanish

Let $f:S^3\rightarrow S^2$ be a smooth map and let $\Omega$ be a volume form on $S^2$. To compute the Hopf invariant $H$ of $f$, we can note that the pullback $f^*\Omega$ is exact and may be written ...
xzd209's user avatar
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2 votes
1 answer
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Let $G$ a lie group and $\varphi:G\times G\to G$ given by $(g,h)\to ghg^{-1}$. what is $(d\varphi)_{(g,h)}(X_g,X_h)$.

Let $G$ a lie group and $\varphi:G\times G\to G$ given by $(g,h)\to ghg^{-1}$. I want to know the derivative of $\varphi$ in the simplest way. I mean, I want to know what is $(d\varphi)_{(g,h)}(X_g,...
Carl's user avatar
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Integration by parts with $|\nabla|:=\sqrt{-\Delta}$

What conditions on $f,g$ do I need to justify the integration by parts $$\int f|\nabla|g\,dx=\int(|\nabla|f)g\,dx.$$From $|\nabla|:=\sqrt{-\Delta}$ we have formally that $|\nabla|$ is a self-adjoint ...
schrodingerscat's user avatar
2 votes
0 answers
42 views

The equivalence relation when constructing the associated bundle

When constructing from a typical fibre $F$, an action of a Lie group $G$ on that fibre and a $G$-principal bundle $P$ over some base space $M$ the associated bundle $P[F]$, one does this by ...
Jannik Pitt's user avatar
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2 votes
1 answer
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A centre of a simple closed planar curve

Let $c: [0,1] \rightarrow \mathbb{R}^2$ be a continuous planar curve such that $c(0)=c(1)$. I was wondering if there is a way to find something like a "centre of mass" of planar curve $c$. ...
Anay Jain's user avatar
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Image of a $C^1$ manifold

Let $F: \mathbb{R}^n \to \mathbb{R}^{n+k}$ be a $C^1$ map with $n, l >0$. It's well know that $F(\mathbb{R}^n)$ is not necessarily a $C^1$ manifold of $\mathbb{R}^{n+l}$. Then what about the ...
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1 vote
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+150

Question about the index of two elliptic operators over a 4-dimensional Riemannian manifold

Let $X$ be a compact Riemannian 4-manifold, $P$ a principal $G$-bundle over $X$, and $\mathfrak{g}$ be its adjoint bundle. Let $\omega$ be a self-dual conneciton on $P$ (i.e. its curvature $\Omega \in ...
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Seeking justification for assumptions in definition of volume entropy of Riemannian manifold [closed]

I am confused about the definition of volume entropy given in the link here: https://en.wikipedia.org/wiki/Volume_entropy Namely, I have two questions: why is the manifold assumed to be compact, and ...
David Schmidt's user avatar
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11 views

Counterexamples to show that "$p < q ⇒ t^{±}(p) < t^{±}(q)$", and "$t^{±}$ are continuous" are not true

About this proposition For a general spacetime $(M, g)$ the volume functions $t^{±}$ (a) $p < q ⇒ t^{±}(p) \le t^{±}(q)$, (b) $t^{±}$ are upper/lower semicontinuous. What counterexamples could I ...
some_math_guy's user avatar
1 vote
1 answer
38 views

$\frac {\partial }{\partial z^{j}}$ and $\frac {\partial }{\partial {\bar {z}}^{j}}$ form bases for complex tangent bundles

Let $M$ be a complex manifold, then there exists a canonical almost complex structure $J:TM \to TM$, where $TM$ is the real rank $2n$ vector bundle on $M$. After complexifying $TM_\Bbb C := TM\otimes \...
Rene's user avatar
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0 answers
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Is a vector field applied to a function a vector or a function? [closed]

Given the covariant derivative of a 1-form $\eta$: $$\nabla \eta (X,Y)=X(\eta(Y))-\eta(\nabla_X Y)$$ Where $X,Y \in \chi(M)$ Which is the rank of the tensor $\nabla \eta$? My thought was that given $\...
Guillermo Fuentes Morales's user avatar
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For a proper Lie group $G$ acting on a complete Riemannian manifold $M$ by isometry, can we find $p'$ close to $p\in M, d(p,p')>d_{M/G}([p], [p'])?$

Let $(M,g)$ be a complete Riemannian manifold, and the Lie group $G$ with $1 \le dim(G) < dim(M)$ acts on $M$ isometrically and properly. Using the metric on $M,$ we define the metric on the ...
Learning Math's user avatar
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1 answer
31 views

Why is this local chart on a fibre bundle $(E, M, \pi, L)$ compatible with the given smooth structure on $E$?

In these lecture notes, at Remark 16.5, Merry states: ''Suppose $(W, \varepsilon)$ is a bundle chart on $E$. Let $(U, x)$ and $(V, y)$ be (manifold) charts on $M$ and $L$ respectively with $W \subset ...
Dave's user avatar
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2 votes
0 answers
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How do I show that the region {r > 2m} of the Schwarzschild spacetime is globally hyperbolic?

How do I show that the region {r > 2m} of the Schwarzschild spacetime is globally hyperbolic? I am just starting with this so I don't really know how to lay out this arguments. I consulted Beem's ...
darkside's user avatar
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-1 votes
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What does it mean if we call a vector field holonomic? [closed]

I would like to know a simple and intuitive way to understand "the vector bundle is holonomic". If it is known that two vector fields X and Y are holonomic, then is it true that their direct ...
lllka's user avatar
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0 answers
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Projection from product of submanifolds is smooth

Consider the following: Let $X_1 \subseteq \mathbb{R}^n$, and $X_2 \subseteq \mathbb{R}^m$ be submanifolds of dimension $k$ and $l$. I have already shown that $X_1 \times X_2 $ is also a submanifold. ...
Philip's user avatar
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1 vote
0 answers
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Any smooth compact 2-dimensional submanifold $S\subset\mathbb{R}^3$ is orientable

A manifold $M$ is orientable if the bundle of antisymmetric $n$-linear forms $\Lambda^n(M)$ is trivial. Equivalently, $\Lambda^n(M)$ admits a nowhere vanishing section. I suppose we must assume that $...
Daniel's user avatar
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Incomplete definition of smooth map between manifolds?

I'm reading Liang's book on Differential Geometry and in that the definition of a smooth map b/w two smooth manifolds is this: Let $M,N$ be two smooth manifolds with dimensions $m,n$ respectively. Let ...
user9343456's user avatar
2 votes
0 answers
50 views

Riemannian structure on the Heisenberg group.

Currently I am reading about Heisenberg Group. And I understand that this group is one of the simplest examples of sub-Riemannian manifolds. I have read a lot about it structure, geodesics and e.t.c. ...
Tat-iva's user avatar
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1 vote
0 answers
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Doubts about alternative definition to connection in differential geometry

We consider a variety M and coordinate bases of the tangent space ${∂_i}^N_{i=1}$, and of the tangent space ${dx^i}^N_{i=1}$. We consider the following definition of connection ∇ on the variety, as an ...
Guillermo Fuentes Morales's user avatar
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0 answers
25 views

Tensors for coordinate transformation

Given a set of matrices $G=\{M_i\}_i^n$ that span some matrix algebra. Now, we are given a linear transformation T acting on this algebra as $T(M_j)= \sum_G U_i^* M_j U_i$. Similarly to usual linear ...
relativeentropy's user avatar
4 votes
0 answers
65 views

Two definitons of a pullback of a differential form

Let $f: X \to Y$ be a morphism of spaces with admitting differential forms (e.g. real manifold, complex manifold, smooth algebraic variety, schemes). Let $\Omega^n_Y$ denote the sheaf of $n$-forms on $...
CJ Dowd's user avatar
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1 vote
1 answer
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Submanifold of matrix space

One can identify the space $M_{\mathbb{R}}(n,n)$ of real $n \times n$ matrices as $\mathbb{R}^{n^2}$. Consider the subset $S:=\{ A \in M_{\mathbb{R}}(n,n) : det(A)=1 \}$ and show it is a smooth ...
Philip's user avatar
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0 answers
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How to compute the first neumann eigenvalue of a spherical lune on the sphere?

Question: What is the first Neumann eigenvalue of Lune $L_{\beta}$ where $\beta$ denotes the opening angle of the Lune on the Sphere $S^2?$ Problem Setup: Parametrize the Lune in geodesic polar ...
Student's user avatar
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