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 ...

learn more… | top users | synonyms (1)

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
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 ...
2
votes
2answers
51 views

Representable homology classes on smooth manifolds

Let $X$ be a closed (compact without boundary) smooth manifold. We can consider its singular homology $H_*(X,\mathbb{Z})$. Let $H_{k}(X,\mathbb{Z})$ be the $k$-th singular homology group of $X$ and ...
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 ...
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 ...
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. ...
0
votes
1answer
29 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
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 ...
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 ...
-3
votes
0answers
8 views

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

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
0answers
30 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
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
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
0answers
49 views

Can you comb the hair on a 4-dimensional coconut?

It is well-known that you can't equip the surface of the unit sphere with a singularity-free coordinate system. Physicists have called this theorem (which is important for the theory of black holes) ...
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 ...
3
votes
2answers
26 views

Confusion about covariant derivative in $\mathbb R^n$

The Levi-civita connection on $\mathbb R^n$ corresponds to the usual directional derivative. In this sense I expect the following to hold: $$ ...
3
votes
1answer
39 views

Question about two ways to induce an inner product on $S^2V$

$\newcommand{\til}{\tilde}$ Let $(V,g)$ be an $n$-dimensional inner product space, and let $S^2V^*$ be the symmetric algebra. I am familiar with a natural way to endow $S^2V^*$ with an inner product ...
1
vote
1answer
31 views

how to change metric variables

The metric on unit sphere is given by: $$g_{ij}=\begin{pmatrix}1 & 0 \\0 & \sin^2{\theta }\end{pmatrix}.$$ The laplacian beltrami operator in $\theta ,\phi$ is $$\Delta ...
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 ...
0
votes
0answers
15 views

Locally defined quantities conserved over a surface

Let "local quantity" mean a quantity that can be defined at any point on a smooth surface embedded in $\mathbb{R}^3$, by a vanishingly small neighborhood of the surface around that point. Let ...
1
vote
3answers
26 views

Simple question about geodesic in Riemann manifold

I was read my book of Riemannian Geometry and the book says the follow: " A parameterized curve $\gamma:I\to M$ is a geodesic in $t_{0}\in I$ if $\dfrac{D}{dt}\left(\dfrac{d\gamma}{dt}\right)=0$ in ...
1
vote
0answers
9 views

Sum two nearest function of two class are the nearest function of the sum class

Suppose $x,\mu:[0,1]\rightarrow \mathbb{R^2}$ two smooth function and $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma (0) = 0, \gamma (1) = 1, \gamma$ is a diffeomorphism $\}$. Here $\Gamma$ ...
0
votes
1answer
19 views

what are the real meaning of exact differential form and closed differential form?

I know that a differential k-form $\omega$ on $U$ is closed if $d\omega=0$, and that is exact so there is a (k-1)-form $\tau$ on $U$ that $\omega=d\tau$. But I don't know real meaning of these ...
0
votes
2answers
64 views

Volume swept by a polygon when it “slides” along a given curve

Given a parametric curve $\mathbf{r}\left(s\right)$, where the parameter $s$ is the length of the curve, lets define a plane $\Pi$ perpendicular to the curve at a specific $s=s_1$. Lets also define a ...
3
votes
2answers
62 views

Calculating the differential of the inverse of matrix exp?

Let $A(t)$ and $B(t)$ be two matrix-valued smooth function satisfying the equation, $B(t) = e^{A(t)}$. I need to express $\frac{dA(t)}{dt}$ in terms of $B(t)$. I know that there is a formula of ...
1
vote
0answers
38 views

A question about Levi-Civita connection and curvature over 3 manifold

Give a 3-manifold M and Riemannian metric $g$, denote $A$ as the Levi-Civita connection on 3-manifold M corresponds to the metric $g$. Denote the curvature of $A$ as $F_A$, choose three bases ...
1
vote
1answer
42 views

About cotangent bundle

$$\text{T}^*U\to\varphi(U)\times(\mathbb{R}^m)^*,\space(x,\lambda)\mapsto\left(\varphi(x),(D_{\varphi(x)}\varphi^{-1})^*(\lambda)\right)$$ What does $D_{\varphi(x)}\varphi^{-1}$ mean? Because I know ...
1
vote
1answer
57 views

Two curves with Same curvature but are not isometric.

question: Give an example of two space curves with the same curvature but are not isometric to each other(there is no isometry between them). I am using the Elementary differential geometry book by ...
1
vote
1answer
29 views

Differential of the multiplication and inverse maps on a Lie group

I'm trying to solve the following two problems: Let $G$ be a Lie group with multiplication map $\mu\colon G\times G\to G$, and let $\ell_a\colon G\to G$ and $r_b\colon G\to G$ be left and right ...
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 ...
1
vote
1answer
30 views

Why does $(X_a,Y_b)(f\circ \mu(a,b)) = X_a(f\circ R_b(a)) + Y_b(f\circ L_a(b))$?

On the first page of these notes: if $\mu\colon G\times G\to G$ is the multiplication map on a Lie group $G$, then given a point $(a,b)\in G\times G$ and letting $R_b$ and $L_a$ denote ...
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 ...
4
votes
0answers
24 views

References for actions of infinite-dimensional Banach-Lie groups on infinite-dimensional Banach manifolds

I am starting to study infinite-dimensional manifolds, specifically, Banach manifolds. I found some interesting introductory texts in which the mathematical background is developed with some detail. ...
0
votes
0answers
16 views

Existence of a solution for a differential equation on a submanifold

Let $S$ be a $m$ dimensionnal submanifold of $\mathbb{R}^N$ and $X$ a tangent vector field to $S$ with regularity $C^{\infty}$. We assume that $X(p_0) \neq 0$ for a gven $p_0 \in S$. Show that ...
3
votes
0answers
24 views

Equivalence of parallel transport, connection and covariant derivative

Here by connection I mean the horizontal distribution. I hear about these three notions are equivalent, i.e. given one we can recover another. In the textbook of Riemannian geometry I have read, ...
2
votes
2answers
76 views

Is the fixed point set of an action a submanifold?

Let $M$ be a differentiable manifold, and $G$ a Lie group acting smoothly on $M$. Under which condition - if any - is the set of fixed points of the action a submanifold of $M$? My thoughts so far: ...
0
votes
0answers
26 views

Let $r(s)$ be a regular closed curve that is in the sphere $S^2$. Prove that $\int_{\gamma}\tau(s)ds=0$ $\gamma$-map of this curve, $\tau$ torsion

Let $r(s)$ be a regular closed curve that is in the sphere $S^2$. Prove that $$\int_{\gamma}\tau(s)ds=0$$ $\gamma$-map of this curve, $\tau$ the function of torsion of this curve. Every year this ...
3
votes
1answer
26 views

Fréchet Topology on $C^\infty(M)$

In the Fréchet space wikipedia article, in the "Examples" section, it is stated that the space of smooth functions $C^\infty(M)$ on a compact smooth manifold $M$ can be made into a Fréchet space "by ...
0
votes
1answer
61 views

if M is compact and N is connected, then M=N …?

Let M and N be surfaces in $R^3$ such that M is contained in N. If M is compact and N is connected, prove that M=N. ================================= I thought intuitively the compactness means ...
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 ...
0
votes
1answer
36 views

Exterior derivative of a two-form with conditions

With the use of this formula $$d\omega(X_1, \dots, X_{r+1}) = ...
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 ...
0
votes
0answers
17 views

Isometry between two planes(differential geometry)

The following exercise problem is contained in O'neill's text. (a) is directly derived from the computations with the definition of the isometry. However, there is something wrong in (b). To ...
0
votes
1answer
22 views

“Sliding” plane in a given curve

Given a parametric curve $\mathbf{r}\left(s\right)$, where the parameter $s$ is the lenghth of the curve, lets define a plane $\Pi$ perpendicular to the curve at a specific $s=s_1$. Lets also define ...
4
votes
1answer
38 views

Differentiability of an action of the group of invertible elements of a $C^{*}$-algebra $\mathcal{A}$ on the dual of $\mathcal{A}$

I am studying the actions of Banach-Lie groups on Banach manifolds, and I am not able to concretely evaluate the differentiability properties of a specific action. Let $\mathcal{A}$ be a unital ...
0
votes
0answers
30 views

Parallel translation along a self intersecting curve

Let $c:I\to M$ be a self-intersecting on manifold $M$. How can one define the parallel translation of the tangent vector $\dot{c}(t_0)$ along $c$? Is the parallel translation along $c$ well-defined?? ...
0
votes
0answers
24 views

Curl by dual: need help with this example

I came across this old thread here: "... Actually, there is a generalization of curl to any dimension. If you have a vector field, you can take its dual. So, the dual of $4 i + 2 j + 3 k$ would just ...
1
vote
1answer
16 views

Orientable on almost complex manifold

I have troubles trying to prove almost complex two-dimensional manifold is orientable. Let I is complex structure on two-dimensional manifold M. Fix a basic $X_1,IX_1$ in each $T_xM$. Easy to see ...
1
vote
0answers
19 views

Discrepancy between line integral over scalar field and line integral over vector field

There is a discrepancy between the line integral over a scalar field and the line integral over a vector field that is bothering me: Say $\gamma$ is a smooth curve. If $\gamma : \mathbb R\to \mathbb ...