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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Relation between Aut(G) and Aut(g)

Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$. We know that when $G$ is simply connected, $\mathrm{Aut}(G)=\mathrm{Aut}(\mathfrak{g})$ (this should follow from the fact that we can ...
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54 views

What are the elements in $\Gamma(\Lambda^2 TM)$?

In the lecture notes, Proposition 1.19 on page 9, it is said that on every Poisson manifold there is a unique bivector field $\Pi \in \Gamma(\Lambda^2 TM)$ such that $$ \{f, g\} = \langle \Pi, df ...
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70 views

Any real analytic Frobenius theorem used in the proof of integrable almost complex manifolds to arise from complex manifolds?

The reference book is S.S.Chern's Complex Manifolds Without Potential Theory. It's used in integrability condition for an almost complex structure to arise from a complex structure. It's claimed that ...
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235 views

Rank Theorem proof

Let $\phi: M \to N$ be an immersion from smooth manifold $M^m$ into $N^n$ ($\dim M = m$ and $\dim N = n$). Prove there exists smooth charts $(U,h)$ in $M$ with $p \in U$, $h(p) = 0$, and $(V,g)$ in ...
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108 views

One-forms in differentiable manifolds and differentials in calculus

Suppose that we have this metric and want to find null paths: $$ds^2=-dt^2+dx^2$$ We can easily treat $dt$ and $dx$ "like" differentials in calculus and obtain for $ds=0$ $$dx=\pm dt \to x=\pm t$$ ...
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78 views

Arc Length and Differential Forms

Suppose $\gamma$ is circle in $\mathbb{R}^3$ defined by coordinates $\begin{pmatrix}r\cos\theta\\r\sin\theta\\0\end{pmatrix}$, and function $F: \gamma \rightarrow \mathbb{R}^3$ is defined by ...
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94 views

A property processed by a special vector field.

Let $Y$ be a vector field on $\mathbb R^n$ (or any Riemannian manifold $(M, g)$). When will we know that $$Y =\nabla_X X$$ for some other vector fields $X$? Or more precisely, if $Y$ does satisfy ...
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38 views

Coordinate dependence of the volume of parallelotope

It is well known that for $n$ vectors $v_1, \ldots, v_n$ in $\mathbb R^n$, the determinant of the matrix $A = (v_1 \ldots v_n)$ [i.e. with the vectors as columns] is related to the volume of the ...
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0answers
77 views

Lie Bracket of vector fields on Lie group

Let $H$ be a Lie group and $\mathfrak{h}$ its Lie algebra. Given a smooth function $v: H \to \mathfrak{h}$, define the vector field $\bar{v} : H \to TH$, $h \mapsto d(R_{h})_{e} v(h)$, where $R_{h} : ...
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69 views

Distance of subgroup to element in Lie groups

Given a (compact, closed) Lie group $G$ and a (closed) subgroup $H$, what is the distance of the identity to $Hg$ (or $gH$), where $g\in G$ and $Hg$ denotes the orbit under left-multiplication? The ...
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53 views

Constructing Manifolds: Submersion

Given a smooth manifold $M$ and a topological space $N$. Consider a local homeomorphism $F:M\to N$ with $\mathrm{im} F=N$. Then one can turn the target space into a smooth manifold via: ...
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80 views

Atlas on product manifold

If {$(U_\alpha ,f_\alpha )$} and {$(V_i,y_i)$} are $C^\infty$ atlases for the manifolds $M$ and $N$ of dimensions $m$ and $n$, respectively, then the collection {$(U_\alpha \times V_i,f_\alpha \times ...
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2answers
60 views

$(n-1)$-alternative tensor on E are decomposable

$E$ is a real vector space with dimension $n$ and $E^*$ is dual space of $E$. Assume $\alpha \in Λ^{n-1}(E)$ Show that there exists $\alpha_1,\alpha_2,...,\alpha_{n-1} \in E^*$ such that ...
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2answers
43 views

Trivialization of $T\mathbb S^1$

It appears to me that $\mathrm{d}/\mathrm{d}\theta$ is a global frame on $T\mathbb S^1$ (geometrically). However, since $\mathrm{d}/\mathrm{d}\theta$ is defined as the pointwise pushforward of the ...
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137 views

double Hodge star operator

$*$ is the Hodge star operator acting on differential $k$-forms on $\mathbb{R}^{n}$ as below: $$*:Λ^k(\mathbb{R}^{n}) \to Λ^{n-k}(\mathbb{R}^{n})$$ $$\alpha \wedge (\star \beta) = \langle ...
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1answer
92 views

Is every smooth $\mathbb{R}$-variety isomorphic to an affine variety?

I sadly don't know anything about formal GAGA yet, but I am at least trying to follow my intuition as often as possible. In differential geometry we know that we can embedd every smooth ...
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78 views

Distance function under diffeomorphism of manifolds

community, I am concerned with measuring distances of systems under diffeomorphisms. Concrety, I consider a smooth diffeomorphism $\varphi: M\rightarrow N$ from the smooth differentiable manifold ...
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1answer
60 views

What does $Du$ mean in a differential equation?

I'm very interested in the following work: http://maths-people.anu.edu.au/~andrews/HSU_Survey141105.pdf . Unfortunately, the author uses (in this and other papers I'm interested in) the notation $Du$. ...
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465 views

metric in the Wasserstein space of gaussian measures

I am reading the paper "Wasserstein Geometry of Gaussian measures" by Asuka Takatsu (section 3 is of interest to me) and I have difficulties understanding how the metric is used. In particular, I am ...
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94 views

Submersion surjective on the complex projective space $\mathbb{C}P^1$.

If $S^3=\{ (z_1,z_2)\in\mathbb{C}^2\mid \vert z_1\vert^2+\vert z_2\vert^2=1\}$ and $\pi:S³\rightarrow\mathbb{C}P^1$ for $(z_1,z_2)\mapsto [(z_1,z_2)]$ since $[(z_1,z_2)]=\{ ...
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3answers
231 views

Definition of the hessian as a bilinear functional on the tangent space

In Milnor's Morse Theory, the Hessian of a smooth function $f : M \to \mathbb R$ defined on a manifold $M$ at a critical point $p$ is the bilinear functional on $T_p M$ defined as follows: ...
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62 views

Do homotopic transition functions define isomorphic bundles (on smooth manifolds)?

It is fairly known that given a cover $\{U_i\}_{i \in \mathbb{N}}$ of some smooth manifold $M$ together with smooth transition functions $g_{\alpha \beta} \colon U_{\alpha} \cap U_{\beta} \rightarrow ...
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125 views

The properness of a submersion

Let $M$ and $N$ be two differential manifolds and there is a surjective submersion $f$ from $M$ to $N$ such that $f^{-1}(p)$ is compact and connected for any $p$ on $N$. Can we conclude that $f$ is ...
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1answer
108 views

Sectional Curvature of Paraboloid

I seem to have made a mistake while doing the simple exercises of calculating 2D sectional curvature of paraboloid $z=\frac a2 (x^2+y^2)$. I used polar coordinates to do this; ...
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78 views

Spaces of constant curvature

Can someone please provide a reference for the theorem that states that, up to isometry, there are only three isotropic spaces of constant curvature, E^n, S^n and H^n, in any dimension.
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101 views

Period of a mechanical system

Im trying to solve the following problem. Consider $\mathbb{R}^{2}$ with coordinates $(x,y)$. Let $H$ be a smooth function on $\mathbb{R}^{2}$. Also, consider the Hamilton equations: ...
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77 views

computation on hyper surface $z=x^2+y^2$

I have problem with following exercise Consider the hypersurface $M$ parametrized by $z=x^2+y^2$. Endow this with the Riemannian metric induced from the $\mathbb{R}^3$. Compute the sectional ...
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1answer
140 views

Riemann sphere, metric derivation-Completed

I have been calculated Riemann sphere, but i got stuck with calculating its metric. Consider complex plane $\mathbf{C}$ and its point $\zeta=\xi+i\eta$. And consider a point in $S^2 / (0,0,1)$ which ...
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86 views

What does $e^{\mu}$ mean for a measure $\mu$?

I have seen the notation $\int_M fe^{\mu}$ in some geometry books and I cannot even guess what $e^{\mu}$ might mean for a measure/form $\mu$ on the (symplectic) manifold $M$. Any clarifications are ...
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1answer
38 views

First part of the proof that $F^*d\beta=dF^*\beta$

Where has the $dy^j$ gone in the highlighted equation? I would have thought the highlighted equation should be $\displaystyle (F^*dg)(x) = \frac{\partial F^j}{\partial x^i}(x)\frac{\partial ...
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1answer
44 views

constant positive K surface

Hilbert's Theorem states that there exists no complete analytic (class Cω) regular surface in $R^3$ of constant negative Gaussian curvature K. For positive Gaussian curvature also when the sphere and ...
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1answer
74 views

geodesics, covering map and its lift

Followings are given problems. Let $f:(M,g)\rightarrow (N,h)$ a covering map that is a local isometry, and let $p\in M$. If $\gamma:[0,1] \rightarrow N$ is a geodesic such that ...
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1answer
112 views

Diffeomorphism between the Grassmannian manifolds $\mathbf{Gr}(n,k)$ and $\mathbf{Gr}(n,n-k)$.

This seems to be a common exercise question, however I am having trouble with it. The hint is to use a map that associates the k-plane to its orthogonal complement. But I have not been able to show ...
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1answer
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What is the metric tensor on the n-sphere (hypersphere)?

I am considering the unit sphere (but an extension to one of radius $r$ would be appreciated) centered at the origin. Any coordinate system will do, though the standard angular one (with 1 radial and ...
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votes
1answer
41 views

Component formula for pullback of one forms

Should it not be $\Big(F'(x)v \Big)^j = \frac{\partial F^j}{\partial x^k }(x)v^k$? Then also how is $F^*dy^j = \frac{\partial F^j}{\partial x^i}dx^i$ derived? I cannot what $\beta_j$ has been ...
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1answer
187 views

How to calculate differential forms on $S^2$ parameterized by stereographic projection?

Suppose that we have the stereographic mapping $\varphi: \mathbb{R}^2\to M$ where $M=S^2-\{(0,0,1)\}$. I've already found that the stereographic parametrization of $S^2-\{(0,0,1)\}$ is given by: ...
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1answer
40 views

Conversion of polar coordinate differential 1-forms to xy-plane

I am new to differential geometry (and StackExchange!) and am having some trouble with the conversion of the polar differential one-forms: $dr$ and $d \theta $. How do I express these in terms of ...
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1answer
72 views

Two regions on sphere with same area

Let $S$ be a regular surface in $\mathbb R^3$ homeomorphic to sphere.Let $f$ be a simple closed geodesic in $S$, and let $A$ and $B$ be the regions of $S$ which have $f$ as a common boundary. be the ...
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pontriyagin class of quaternionic vector bundle

Let $\xi^{\mathbb{H}}$ be a quaternionic vector bundle over $X$. How to define the Pontriyagin class of $\xi^{\mathbb{H}}$ efficiently? Of course we can let $(\xi^{\mathbb{H}})_{\mathbb{R}}$ be the ...
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1answer
55 views

function of class C ^ 1 on manifolds

Let $M$ be a differentiable manifold with finite dimension $ m $. Let $ f:M\rightarrow M $ a function of class $C^1$. I have a doubt about what this implies (1) or (2): $x \in M \rightarrow D_xf ...
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frustrating experience about differential geometry

I am felling rather frustrated now, after taking a long time to study differential geometry, but with little progress... Indeed my major is mainly numerical analysis. I am studying modern geometry, ...
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72 views

Smooth map on submanifold

Is the following true? Let $M$ be a differential manifold and $f : M \to M$ be a smooth map. If $N$ is a submanifold of $M$ and $f(N)\subset N$ then the restriction $f|_N : N \to N$ is smooth.
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Proving that the change of parameters is differentiable.

Let $M \subset \Bbb R^3$ be a regular surface, and ${\bf p} \in M$. Let ${\bf x} : U \subset \Bbb R^2 \to M$ and $\overline{{\bf x}}:\overline{U} \subset \Bbb R^2 \to M$ be parametrizations at ${\bf ...
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105 views

Creating topological spaces with portals

I'm trying to rigorously describe an object that I'm calling a "portal". The situation is easiest to describe in two dimension. I start with a line segment $pq$ in $\mathbb{R}^2$. I want to remove ...
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1answer
51 views

Parallel Transporting a vector

I want to parallel transport a vector $V^{\mu}$ with the initial condition $V^{\mu} = (V^{\theta},V^{\phi}) = (1,0)$ along a closed curve parameterized by $ \lambda \in [0,1]$ and determine the ...
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1answer
82 views

Is rectangle manifold with boundary

Is a closed rectangle a 2d manifold with boundary? It seems like the corners don't have neighborhoods homeomorphic to the Euclidean half space?
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509 views

Prove the sphere is orientable

Is there an easy way to show that the sphere is orientable other then using stereograohic projection. I am preferably looking for something derived from a basic theorem in elementary geometry with ...
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1answer
128 views

Meaning of the adjoint representation of a Lie group

The adjoint representaion of $G$ is a homomorphism $ad_{a}:g \rightarrow aga^{-1}$, $a,g \in G$, what is the meaning of this? Now if we identify $T_{e}G$ with $\mathfrak{g}$ we have the adjoint map ...
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1answer
79 views

How to calculate curl curl E using differential forms?

We can calculate $\mathbf{curl}\,\mathbf{E}$ by $d(E^1dx_1+E^2dx_2+E^3dx_3)$. But how to calculate $\mathbf{curl}\,\mathbf{curl}\,\mathbf{E}$ using differential forms? I know the first ...
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132 views

Non-vanishing differential forms

Let $M$ be a differentiable manifold of dimension $n$. If the tangent bundle is trivial, then the cotangent bundle is trivial, and so are its exterior powers. In other words, on a parallelizable ...