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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6
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1answer
285 views

How to prove the holonomy group is preserved under Ricci flow?

I've heard that on a Kähler manifold $(M,g_0)$, if you evolve the metric $g$ by Ricci flow $\partial g_{ij}(t)/\partial t=-2R_{ij}$, and $g(0)=g_0$, then you always have $g(t)$ is a Kähler metric on ...
2
votes
1answer
72 views

Exercise about Lagrangian submanifolds

I am trying to solve the following exercise: Let $(M,\omega)$ be a symplectic manifold and $L$ a compact Lagrangian submanifold such that $H^{1}(L)=0$. Let $\{L_{t}\}_{t\in(-1,1)}$ be a smooth family ...
1
vote
1answer
314 views

Gaussian Curvature

Can anyone see the connection between the Gaussian curvature of the ellipsoid $x^2+y^2+az^2-1=0$ where $a>0$ and the integral $\int_0^1  {1\over (1+(a-1)w^2)^{3\over 2}}dw$? I am guessing ...
0
votes
1answer
16 views

Proof hyperbolic surfaces have two asymptotic directions

I have seen it stated that: For every point in a surface of negative Gaussian curvature, there are exactly two asymptotic directions, i.e ones in which the normal curvature is zero. How can this ...
3
votes
1answer
23 views

$S^1$ acting on $SO(n+1)/SO(n-1)$ by translations

I'm ready right now in a paper, that $S^1$ acts on $SO(n+1)/SO(n-1)$ by right translations. I thought that a Liegroup $G$ acting by right translations, means that we have a right action $\varphi ...
3
votes
1answer
636 views

Understanding Lipschitz domain

Here is the definition of Lipschitz domain given by Wikipedia. Let n ∈ N, and let Ω be an open subset of Rn. Let ∂Ω denote the boundary of Ω. Then Ω is said to have Lipschitz boundary, and is called ...
0
votes
1answer
17 views

Question on the support of a vector fields, and on the set of critical points

Let $M$ be a smooth differential manifold, and let $X\in\mathfrak{X}(M)$ be a smooth vector field. The support $supp(X)$ of $X$ is defined as the closure, in $M$, of the set $A(X):=\{m\in M: X(m)\neq ...
2
votes
2answers
130 views

How prove this $S_{\Delta ABC}\ge\frac{3\sqrt{3}}{4\pi}$

There is convex body $T$ (with the area is $1$), show that there is a triangle $\Delta ABC$, such $A,B,C\in T$, and $$S_{\Delta ABC}\ge\dfrac{3\sqrt{3}}{4\pi}$$ This problem is from China ...
-1
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0answers
34 views

Solvable group which is not Virtually nilpotent

What is a example of Solvable group which is not Virtually nilpotent(does not have any nilpotent subgroup of finite index)?
0
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0answers
24 views

Full definition of Rough Laplacian and induced formal adjoint of covariant derivation?

Can everybody give a good reference for full definition of Rough Laplacian of tensor field and induced formal adjoint of covariant derivation on a riemannian manifold? I find some equivalent ...
0
votes
0answers
38 views

Definition of Regular Point

In "An Introduction to Manifolds (second edition)" by Louring W.Tu, regular point is defined as: Definition 8.22. A point p in N is a critical point of F if the differential $F_{∗,p}: T_pN ...
0
votes
1answer
23 views

Compute the associated induced Lie algebra action $\text{d}\pi$

Let $G=\mathrm{SL}_2(\mathbb{C})$ and consider the action of $G$ on the space of smooth functions on column vectors $\mathbb{C^2}$ given by $\big(\pi(g)\phi\big)(v)=\phi\left({g^\top}\,v\right)$ for ...
0
votes
0answers
54 views

Why such coordinates are still called “isothermal” in the Lorentz case?

We know that in a two dimensional Riemannian manifold there always is, at least locally, a coordinate system in which $g_{ij} = \lambda^2 \delta_{ij}$ - such coordinates are called isothermal. There ...
2
votes
1answer
541 views

Gaussian curvature $K$ of of orthogonal parametrization $X$

Let $X$ be an orthogonal parametrization of some surface $S$. Prove that the Gaussian curvature $K = - \frac{1}{2 \sqrt{E G}} ((\frac{E_{v}}{\sqrt{E G}})_{v} + (\frac{G_{u}}{\sqrt{E G}})_{u})$, where ...
0
votes
3answers
44 views

Reference request: integration of *one*-forms along curves on a differentiable manifold.

Could somebody please direct me to a book/lecture notes with an introduction to integration of one-forms along curves in a differentiable/Riemannian manifold -- preferably leaning more towards ...
0
votes
1answer
21 views

Line of Curvature/geodesic is a plane

Let me preface this question with: I have read the related and almost exact questions previously posted. Due to lack of points I cannot comment additional questions on those posts. I have also made ...
1
vote
0answers
36 views

An intuitive question about the metric $d(\cdot,\cdot)$ on a complete Riemannian manifold

Let $(M,g)$ be a complete Riemannian manifold. Suppose $\gamma_1,\gamma_2:[0,1] \to M$ are two different segments(i.e.minimizing geodesics) from $p$ to $x=\exp_p(v)$. Suppose further that ...
1
vote
1answer
83 views

Exercise about cotangent bundle

I am trying to solve the following exercise: Let $Q$ be a manifold, $T^{*}Q$ its cotangent bundle and $\pi:T^{*}Q\rightarrow Q$ the bundle map. Let $\omega$ be the standard symplectic form on ...
1
vote
1answer
46 views

Why is this contradiction occurs?

The following text is from the book pifferential Geometry of Curves and Surfaces by M. do Carmo : $\quad $ Let $\alpha:I\to\Bbb R^3$ be a curve parametrized by arc length without singular points ...
1
vote
1answer
9 views

Are segment domains closed?

Let $M$ be a complete Riemannian manifold. Its segment domain is defined by: $$ \mathbf{seg}(p)= \{v\in T_pM: \exp_p(tv):[0,1] \to M \textit{ is a segment} \ \ \} $$ (Note: "segment" has many ...
0
votes
1answer
30 views

Vector bundle vs Total Space

On page 59 in Lee's "An Introduction to Smooth Manifolds" the author writes, "Let $E$ be a smooth vector bundle over a smooth manifold $M$, with projection $\pi:E\to M$." I thought the vector bundle ...
0
votes
1answer
27 views

Condition on symplectic form: $(d\alpha)^n \neq 0$?

I started to read about contact and symplectic forms and I came across this answer here. It seems to state that the definition of symplectic form is that $d\alpha$ is non-degenerate if and only if ...
2
votes
1answer
58 views

Why is a differential equation a submanifold of a jet bundle?

I'm reading The geometry of jet bundles by D.J Saunders and struggle with the definition of a differential equation 6.2.23. on page 203. First of all, Saunders introduces a differential operator ...
1
vote
1answer
42 views

Local picture of a Riemannian manifold with constant sectional curvature.

Theorem: If a Riemannian n-manifold $(M, g)$ has constant sectional curvature $k=1$, then every point in $M$ has a neighborhood that is isometric to an open subset of the space form $S^n$. (cf. ...
3
votes
0answers
35 views

costruction of brownian motion on sphere?

i am trying to construct a brownian motion on the sphere using the method given in Price and williams paper.$\partial$ represents the SDE of stratonovich type which is converted to ito form in last ...
4
votes
0answers
128 views

Surface area from indicator function

I know that the volume and the surface area of a sphere of radius $R$ are related by a derivative: $$V(R)=\frac{4}{3}\pi R^3$$ $$A(R)=4\pi R^2=\frac{\partial V(R)}{\partial R}$$ I am asking if an ...
1
vote
1answer
46 views

Understanding the definition of a closed manifold

Let $D\subset R^n$ be a bounded domain with smooth boundary. Is $\partial D$ a closed manifold?
3
votes
1answer
62 views

Proof that this is a smooth manifold

Let $F: \mathbb{R}^2 \rightarrow \mathbb{R}^4$ be defined by $F(u,v) = (u+v, uv, u-v, v^3)$ and let $M = F(\mathbb{R}^2)$. Prove that $M$ is a smooth manifold. The proof that my TA posted online was ...
3
votes
2answers
40 views

Non-orientable manifolds and mod 2 homology

I was reading the wonderful book "The wild world of 4-manifolds" by Alexandru Scorpan and I found the following sentence: "We are able to orient $\mathfrak{M}$ (else we only get modulo 2 ...
1
vote
0answers
20 views

Confusion surrounding the Koszul-Malgrange theorem

On a recent project, I ran into something which seemed related to the Koszul-Malgrange theorem. According to nlab, the original statement of the theorem is Theorem 1. (Koszul-Malgrange theorem) ...
1
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1answer
34 views

Isometric action on $S^n$

Let $S^n$ be the n dimensional sphere. For $n=2k+1$ odd, we identify $S^n$ as subset of $\mathbb{C}^{k+1}$. Furthermore we can define the action $$\Psi: S^1 \times S^n \to S^n, (c,(z_0, \dots, z_n)) ...
2
votes
2answers
33 views

Why image of curvature is a Lie subalgebra?

In the red line of picture below, why it is Lie algebra ? $M_{\alpha\beta}$ is the Lie bracket ? But $M_{\alpha\beta}$ is symmetric . Picture below is from the 216 page of this paper. ...
0
votes
0answers
24 views

Proof of Hamilton's strong maximum principle.

As picture below, Why $\forall v\in \text{null}(M_{\alpha\beta}), \nabla_iv\in\text{null}(M_{\alpha\beta})\Rightarrow \text{null}(M_{\alpha\beta}) \text{ is invariant under parallel translation}$ ? ...
1
vote
2answers
50 views

Deriving the round metric

I want to derive the round metric $g=d\theta^{\,2}+\sin\left(\theta\right)^2d\phi^{\,2}$ but I cannot get the correct answer. I know that the metric in cartesian coordinates is $g=dx^2+dy^2$. I've ...
0
votes
0answers
27 views

Rank of curvature operator under Ricci flow.

I think under Ricci flow ,the rank of curvature operator does not change by +1 or -1, it will directly change to full rank or zero rank . I want to write it as term paper, but I don't know whether ...
1
vote
0answers
8 views

Cokernal of the map $B_1 \times B_2 \rightarrow [B_1,B_2]$

I am reading Nakajima's book, Lectures on Hilbert Schemes of Points on Surfaces. In the proof of Theorem 1.9, it needs the following linear algebra fact. Suppose $B_1, B_2$ are two $n \times n$ ...
0
votes
0answers
30 views

Critical points of Exponential Map

How do you find critical points of an exponential map? I am working with a sphere of radius R. I know that an exponential map maps vectors of the tangent plane to a neighborhood of the point Q on the ...
0
votes
0answers
33 views

Does $\#_n S^2 \times S^1$ really admit a map of non-zero degree from $B \times S^1$

In this paper, http://arxiv.org/abs/1202.6302, the authors claim to produce a degree 2 $\pi_1$-surjective map $f$ between $M=S^1 \times \Sigma_2$ and $N=\#_2 S^2 \times S^1$ (the connected sum of two ...
0
votes
1answer
26 views

Global parametrization of submanifolds

Is the following true ? : Let $M$ be a $C^1$-submanifold of $\mathbb{R}^n$ of dimension $k\leq n$ (without boundary). There is a $C^1$-map $f$ on an open subset $U$ of $\mathbb{R}^k$ such that ...
1
vote
1answer
23 views

Tensor field acting on vector fields and covector fields gives a function?

If I have a $(p,q)$ tensor field that acts on $p$ covector fields and $q$ vector fields then does $T\left(X_1,\dots,X_p,Y_1,\dots,Y_q\right)$ return a function $f$ defined on the manifold by ...
0
votes
0answers
7 views

What is orthogonal part of product of two isometries?

Problem says: Prove the general formulas $(GF)_{*}=G_{*}F_{*}$ and $(F^{-1})_{*}=(F_{*})^{-1}$ in the special case where $F$ and $G$ are isometries of $\mathbb{R}^{3}$ To solve it, I ...
0
votes
0answers
16 views

determine curve only by normal direction.

let $\alpha (s) $ be arc-length parametrized curve in $\mathbb{R}^n $. Assume that we know about normal vector $N(s) = \frac{\alpha ''(s)}{| \alpha ''(s) |} $, but we have no other information about ...
0
votes
1answer
28 views

Prove the pullback of the wedge product is the wedge product of the pullbacks.

Let $F:V \rightarrow W$ be a linear map. Show that $F^{\ast}(\omega \wedge \eta)=(F^{\ast}\omega) \wedge (F^{\ast}\eta)$ for all $\omega \in \Lambda^{p}(W) , \eta \in \Lambda^{q}(W)$. Where ...
0
votes
1answer
34 views

Integral of Differential 1-form

Let $\omega$ be the closed $1$-form $\omega = \frac{xdy - ydx}{x^2 + y^2}$ and let $S$ be the unit circle and let $C = \{(x,y) : (x-3)^2 +y^2 = 1\}$. I'm trying to find $\int _S \omega$ and $\int ...
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0answers
8 views

M is set with O as topology defined on it alongwith this smooth atlas and connection is given.

Chart transition maps are given and connection coefficient functions with respect to polar chart are to computed.this is actually a tutorial problem of online light and gravity course at WE-Heraeus ...
1
vote
2answers
62 views

How to find two inequivalent ,but weakly equivalent bundles?

I want to know if I can have an inequivalent but weakly equivalent bundles over the Torus, i.e. if we can find explicitly an example of a bundle map $(\overline{f},f)$ where we require that ...
1
vote
0answers
29 views

Length of a differentiable curve with respect to a Riemannian metric.

Let $X$ be an $n$-dimensional differentiable manifold ($n\ge1$). A Riemannian metric in $X$ is a family $\{g_p\,|\,p\in X\}$, where for all $p\in X$: $g_p:T_pX\times T_pX\to\mathbb{R}$ is an inner ...
0
votes
0answers
17 views

Result involving bundles [duplicate]

I want to know if I can have an inequivalent but weakly equivalent bundles over the Torus, i.e. if we can find explicitly an example of a bundle map $(\overline{f},f)$ where we require that ...
0
votes
1answer
16 views

Explicit non-singular coordinate system for $S^3$

Define a "non-singular" coordinate system on a manifold as a continuous, everywhere differentiable set of coordinates such that the determinant of the metric tensor $g_{\mu\nu}$ is everywhere ...
1
vote
1answer
28 views

Geometric quantization: not understanding the curvature form and Weil's theorem

I am reading a bit about geometric quantization. The texts that I am following are N.M.J. Woodhouse's "Geometric Quantization" and an article from arXiv. I am having troubles understanding the ...