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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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
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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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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 ...
gaoqiang's user avatar
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Given $F: \mathbb R^n \to\mathbb R^m,$ $n < m,$ if $F$ has no critical points in a subset $U,$ then is F necessarily injective in $U?$

Given a map $f:\mathbb R^n \to\mathbb R^m,$ $n<m,$ and a connected subset $U$ of $\mathbb R^n$ where $F$ has no critical points, is this a sufficient condition for $F$ to be injective in $U$? Would ...
gaaaaaaaaaah's user avatar
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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 ...
user302934's user avatar
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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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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
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$\frac {\partial }{\partial z^{j}}$ 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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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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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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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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What does it mean if we call a vector field holonomic?

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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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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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
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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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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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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
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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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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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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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When is $h=d\omega$

Given $h\in \Lambda^{k+1}(M)$, where $M$ is some manifold, how do we know that $h=d\omega$ for some $\omega\in \Lambda^k(M)$? For example, consider $$h=h_1(x^1,x^2,x^3)\,dx^1\wedge dx^2+h_2(x^1,x^2,x^...
Irene's user avatar
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Compute Christoffel symbols of sphere by embedding

In the answer https://math.stackexchange.com/q/4748125 of V. Semeria, Taking $$(y_1,\dots,y_{n+1})=(x_1,\dots,x_n,\sum_{i=1}^{n+1}x_i^2 -R^2)$$ Write $(\vec{e}_1,\dots,\vec{e}_{n+1})$ the canonical ...
Measure32's user avatar
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1 answer
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Length of a regular arc

I have to prove that if two arc are (positive) equivalent, they have the same length. Let $\alpha:[a,b]\to \mathbb{R}^n$ and $\beta:[c,d]\to \mathbb{R}^n$ two differentiable arcs and let $h:[a,b]\to [...
Sigma Algebra's user avatar
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1 answer
32 views

Möbius strip is a smooth submanifold

I want to find a description of the Möbius strip without boundary as a submanifold. To be more specific, what I mean with "description". I know the following proposition: For a subset $M \...
Philip's user avatar
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Algebraic Geometry book for someone interested in Differential Geometry [duplicate]

I am very interested in differential geometry. However, in order to broaden my horizons, I would like to know the connection between algebraic geometry and differential geometry, any recommendations ...
Chengrui Han's user avatar
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1 answer
42 views

Where is the mistake in this proof that 2-form is non-vanishing?

Suppose $F : \mathbb{R}^3 \rightarrow \mathbb{R}$ is $C^{\infty}$ and that $0$ is a regular value, so that $M = F^{-1}(0)$ is a $2$-dimensional submanifold of $\mathbb{R}^3$. The goal is to prove that ...
Jack Ceroni's user avatar
5 votes
1 answer
60 views

Geometry of origami saddle surfaces made of five or six square paper sheets connected around a point

I connected five and six square paper sheets (which are all initially flat and have the same dimensions) using tapes to create two smooth saddle surfaces (see below), but I couldn't figure out the ...
Wayne's user avatar
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3 votes
1 answer
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Concatenation of $f$-related vector fields/tangent vectors

This is probably a highly trivial question, but I just can't wrap my head around it. Let $M,N$ be two manifolds and $f: M \rightarrow N$ smooth. Let further $X^1, X^2$ be two smooth vector fields on $...
welahi's user avatar
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0 answers
27 views

Hodge decomposition on a riemannian manifold for another measure?

Let $M$ be a closed riemannian manifold, we denote $dx$ the canonic volume measure on $M$. If we take $\mu$ a probability measure such that $d\mu = \rho dx$ with $\rho > 0$ we can define $d_\mu^*$ ...
Aymeric Martin's user avatar
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38 views

Connection with constant Christoffel symbols

Assume that $(\mathcal{M},\nabla)$ is a manifold diffeomorphic to $\mathbb{R}^n$ equipped with an affine connection $\nabla$. Assume that there is a chart in which the Christoffel symboles, defined as ...
Chevallier's user avatar
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1 answer
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Dot product of two exterior products and associativity of geometric product?

This is a quick and basic question. I looked online (Wikipedia articles, Wolfram, etc..., and poked inside of Hestenes and Snygg's books, but couldn't easily pull out an answer). I'm going to define ...
Nate's user avatar
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10 views

Fixed a point $p$ and a ray $\gamma$, there is a only ray $\sigma$ from $p$ asymptotic to $\gamma$?

Assume $M$ is a complete, oriented, noncompact simple connected 2-D Riemannian manifold. Let $\gamma:[0,\infty)\to M$ be a ray of $M$ which means for all $t_1,t_2$, $d(\gamma(t_1),\gamma(t_2))=|t_1-...
Lacen's user avatar
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2 votes
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What is a transformation (physical) in precise terms?

I’m reading some quantum field theory from physics (David Tong’s lecture note), but I don’t quite understand what it means to perform some transformation $x \rightarrow x’$. I’m trying to view it in ...
Frozer Clark's user avatar
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+50

Computing the tangent space of the orbit of a gauge group action at a connection

Let $E\to M$ be a smooth real vector bundle, and let $\mathfrak{G}$ be the group of smooth bundle automorphisms. (The Lie algebra of $\mathfrak{G}$ is the space $\Omega^0(\text{End}(E))$.) For a ...
blancket's user avatar
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Interpreting $\int_\gamma f = \int P\ \text{d}x + Q\ \text{d}y$ via differential forms?

I've often seen the expression $\def\g{\gamma}\def\dx{\text{d}x}\def\dy{\text{d}y}\def\td{\text{d}}\def\br{\textbf{r}}\def\dt{\text{d}t}\def\RR{\mathbb{R}}$ $$\int_\g f\ \td\br = \int P\ \dx+Q\ \dy$$ ...
Sam's user avatar
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1 vote
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20 views

The second order covariant derivatives of the first eigenfunction

In the book of Schoen-Yau, "Lectures on Differential Geometry", in Proposition 1.7, page 195, they said that: $\textbf{The second order covariant derivatives of the first eigenfunction $\phi$...
Duc's user avatar
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0 answers
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Height function on the sphere is Morse

I'm trying to prove that the height function on the sphere $\mathbb{S}^2$ is a Morse function. Somehow from my onw calculations, I conclude that it's not Morse. Using the embedding with spherical ...
SiS_sos's user avatar
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23 views

Fisher-Rao distance and Hellinger distance

I have read on a tweet that "The Fisher-Rao distance is the geodesic distance associated to smooth divergences (eg. Kulback-Leibler). Without constraint, it is the Hellinger distance" and I ...
Ramufasa's user avatar
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57 views

Examples of "smooth" sets in $\mathbb{R}^2$, $\mathbb{R}^3$

What is the most simplest straight forward example of a set in $\mathbb{R}^2$ or $\mathbb{R}^3$ which has "smooth" boundary? Where $H(Q_+)=U\cap Q$ needs to be replaced by $H(Q_+)=U\cap \...
Perelman's user avatar
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-1 votes
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Unique Normal Vector for C^1 curves parametrized by arclength

i've been working on a problem for quite a while now and couldn't make any progress. So the basic question is as follows: Consider a closed and injective curve $\gamma:\mathbb{S}^1\to\mathbb{R}^3$ of ...
KingKermit's user avatar
1 vote
0 answers
28 views

Characterizing the complex structure on a non-compact Riemann surface

(This is a crosspost from MathOverflow) A consequence of Torelli's theorem is that a closed Riemann surface $X$ is determined by its period matrix. More precisely, fix $(\alpha_i)$ a basis of $H_1(X, \...
Louis Beaufort's user avatar
1 vote
1 answer
56 views

Confusion about one pushforward calculation on $\mathbb{S}^{n-1}$

Let $\mathbb{S}^{n-1} \subset \mathbb{R}^n$ be the standard sphere and let $$ F \colon \{ (p, r, X) \colon \ p \in \mathbb{S}^{n-1}, \ r > 0, \ X \in T_p \mathbb{S}^{n-1}, \ |X| = 1\} \to \mathbb{S}...
tsnao's user avatar
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1 vote
1 answer
31 views

Two $SO(2)$ connections on a smooth $SO(2)$-vector bundle

Let $L\to M$ be a smooth $SO(2)$-vector bundle. We can identify the Lie algebra $\mathfrak{so}(2)$ with $i\Bbb R$. Suppose $\nabla, \nabla'$ are two $SO(2)$-connections on $L$ such that $\nabla=\nabla'...
blancket's user avatar
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Why $C(g)$ is a representation of $G$?

I'm trying to study on some lecture notes about $G-$structures of Crainic, but I don't understand the following remark: We will encounter several quite complicated vector bundles associated to a $G-$...
Armando Patrizio's user avatar
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0 answers
20 views

Divergence in curvilinear coordinates and Voss-Weyl and covariant basis vs standard normalized basis [duplicate]

I am not very confident with curvilinear coordinates and I saw this answer where the author says that in ordinary polar coordinates, the divergence is: $$\partial_r V^r + \frac{1}{r} V^r + \partial_\...
atapaka's user avatar
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1 answer
53 views

Understanding of connections differential geometry

I have some questions regarding connections and christoffel symbols. The definition of connections im working with is simpler than the more general definition on a vector/tensor bundle, it is the ...
John Doe's user avatar
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1 vote
1 answer
67 views

Why $\phi_{*0}(\frac{d}{dt}|_0)=\frac{d\phi^{\mu}(t)}{dt}|_0\frac{\partial}{\partial g^{\mu}}|_e$?

Let $G$ be a Lie group and $\phi :\mathbb{R}\to G$ be a smooth homommorphism. I know that $\phi_{*0}:T_0 \mathbb{R}\to T_eG$ is a linear map, $T_0 \mathbb{R}=\langle \frac{d}{dt}|_0\rangle $, and $T_e ...
Mahtab's user avatar
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