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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49 views

Constructing $\mathbb{P^n}$ “bundle” with different $n$

Can we find a complex/smooth manifold and map $f\colon X\to\mathbb{C}^3$ such that it is a $\mathbb{P}_\mathbb{C}^m$ bundle over linear subspace $\mathbb{C}^1$ and $\mathbb{P}_\mathbb{C}^n$ bundle ...
5
votes
0answers
108 views

A Quotient of the Euclidean Group

$\newcommand{\euc}{\mathscr I}\newcommand{\R}{\mathbf R}$ Let $\euc(n)$ denote the the Euclidean group $\R^n\rtimes O_n(\R)$. Recall that $\euc(n)$ acts on $\R^n$ as $(\mathbf x, T)\cdot \mathbf ...
5
votes
3answers
187 views

Can a Compact Lie Group have a Non-Compact Lie Subgroup?

Hopefully the title says it all in terms of the question I'm asking. I feel that the answer is no, but I'm not sure why; I don't really know many deep theorems about the structure of lie groups.
1
vote
1answer
60 views

Self-indexing Morse function on a torus which is a height function

A Morse function $f: \Bbb T^2\to [0,2]$ is called self-indexing if $f^{-1}(n)$ is the set of critical points of index $n$. It is relatively easy to see that on any compact manifold, any Morse function ...
16
votes
3answers
2k views

Geometric interpretation of connection forms, torsion forms, curvature forms, etc

I have just begun learning about the connection 1-forms, torsion 2-forms, and curvature 2-forms in the context of Riemannian manifolds. However, I am finding it hard to relate these notions to any ...
0
votes
1answer
39 views

Is $M=\{(x,y,z)\in \mathbb{R}^3: x^3+y^3+z^3=1, z=xy\}$ a manifold of class $C^{\infty}$?

Let $M=\{(x,y,z)\in \mathbb{R}^3: x^3+y^3+z^3=1, z=xy\}$. Is $M$ a manifold of class $C^{\infty}$? I need find a atlas $\{(U_i,\varphi_i)\}_{i\in I}$ with $U_i$ open sets and $\varphi$ ...
0
votes
0answers
35 views

Proof of Theorem 3.2 - Elementary Differential Geometry, O'Neil.

I am going through Elementary Differential Geometry by O'Neil, and I am at Theorem 3.2 on page 151. O'Neil comments that a rigorous proof of this theorem requires the methods of advanced calculus, and ...
1
vote
1answer
38 views

Gradient in terms of first fundamental form

In Do Carmo's Differential Geometry of Curves and Surfaces, I'm having a quite hard time trying to solve Excersise 14 on pages 101-102. He defines the gradient of a differentiable function $f:S\to ...
0
votes
1answer
21 views

Reparametrization Confusion

Why is it true that if $\hat{\gamma}$ is a reparametrization of $\gamma$, then $\gamma$ is a reparametrization of $\hat{\gamma}$? I think I understand, but i'm not sure how to formally show this. A ...
1
vote
0answers
46 views

Hodge theory in general

I know a bit of Hodge theory, and I know that there is an analogue in the symplectic case, where instead of inducing the $\star$-product using the metric we use the symplectic form. Is in true in ...
0
votes
0answers
31 views

Harmonic map and pullback metric

Let $\phi : M \to \mathbb{R}^n$ be a harmonic map, where $M$ is a Riemannian manifold. Let us take coordinates $(u_1, u_2,..., u_n)$ on $\mathbb{R}^n$ and express the Euclidean metric as $g = \Sigma ...
-1
votes
0answers
14 views

Curvature differential geometry [closed]

A curve lies in the interior of a circle with radius r. Tangent of circle at point p is same as that of curve, and curve is regular curve. Want to show that curvature of curve at p is greater than or ...
3
votes
0answers
48 views

Determinant of non-square Jacobian

Suppose I have a 3d solid in ${\bf R}^4$ which can be parametrized by the function $F:W\subset{\bf R}^3\rightarrow{\bf R}^4$. Now suppose I want to calculate the volume of this solid. Then naively I ...
1
vote
0answers
21 views

when can a surface conformally equivalent to the sphere be isometrically immersed?

Given a scalar function $s:S^2\to \mathbb{R}$, and the induced Euclidean metric $g$ on $S^2$, when can the sphere, equipped with metric $e^sg$, be isometrically immersed in $\mathbb{R}^3$? Is there a ...
0
votes
0answers
28 views

Change of coordinates in target space of map

Consider a function $\phi = (\phi_1,....,\phi_n) : \mathbb{R}^m \to \mathbb{R}^n$. Suppose that $\phi_i$ is harmonic for each $i$, that is, $-\Delta \phi_i = 0$. Suppose we change from Cartesian to ...
0
votes
0answers
23 views

Helix terminology

Consider the following helix: Is there a mathematical term for the section highlighted in red? Two that immediately spring to mind are "period" and "cycle". However, my advisor (a differential ...
3
votes
1answer
49 views

Punctured complex projective space

Let $\mathcal{P}\mathbb{C}^{n}$ be the complex projective space of $\mathbb{C}^{n+1}$, and let $B=\{\mathbf{e}_{1},\cdots,\mathbf{e}_{n+1}\}$ be a basis in $\mathbb{C}^{n+1}$. I would like to ...
1
vote
0answers
18 views

When does a non singular integrable differential one-form define a regular foliation?

Let $\mathcal{M}$ be a smooth manifold of dimension $m<+\infty$. Let $\theta$ be a nowhere vanishing (non-singular) differential one-form on $\mathcal{M}$ such that $\theta\wedge d\theta=0$. ...
1
vote
1answer
21 views

Interior product general rule (differential forms)

How is this general form of interior product on forms $$(i_V\omega^{(p)})=\frac{1}{(p-1)!}V^{\mu}\omega_{\mu\mu_1...\mu_{p-1}}dx^{\mu_1}\wedge dx^{\mu_2}\wedge ...\wedge dx^{\mu_{p-1}}$$?
0
votes
1answer
43 views

How does the solution of a differential equation on a manifold yield a map?

In: "A solution $x^μ(λ)$ is a map from $\mathbb{R} → M$": Why is $x^μ(λ)$ considered a map and why does it go from $\mathbb{R} → M$? I can't seem to illustrate this in my mind. In:"If the manifold ...
0
votes
0answers
19 views

Taylor expansion of length of a geodesic via Jacobi field approach

I'm currently reading a paper and am stuck in a proof that is connected with Jacobi fields. The situation as given is the following: I do have two curves $\alpha, \gamma$ starting starting from two ...
3
votes
1answer
23 views

How to indicate the space of sesquilinear forms?

Is there a canonical way to indicate the space of sesquilinear forms? For bilinear forms I can use the (0,2) Tensor space, but since sesquilinear forms are not linear in one variable I don't know how ...
2
votes
0answers
32 views

Parameterization and geodesics of a 3-torus

So I'm thinking about a space exploration game where the primary mechanic is to fly a space ship around the surface of various 4-dimensional surfaces. The way I'd like to render this is by ray casting ...
1
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0answers
26 views

The first eigenvalue for Dirichlet boundary condition positive?

Let $M$ be a compact, n dimensional Riemannian manifold with boundary. Then we know that $W^{1,2}(M)=W^{1,2}_0(M)$, the latter is the completion of $C_0^{\infty}(M)$ function w.r.t $W^{1,2}(M)$-norm. ...
1
vote
1answer
44 views

Angle between position and velocity vectors is constant?

Is there a name for such a curve or can this even happen? I know when the velocity vector, $\mathbf{x'}$, and position vector, $\mathbf{x}$ are always orthogonal $\mathbf{x}(t)$ parametrizes a circle ...
3
votes
1answer
53 views

Unit Normal vs Principal Normal

Here is the problem I am working on: Deduce the equation of the main normal and binormal to the curve: $x=t, y=t^2, z=t^3, t=1.$ I remember from Calc-3 that the binormal is unit tangent $\times$ ...
3
votes
0answers
34 views

$\mathbb{R}^n$ bundle over $B$ being compact and having finite covering dimension implies has finite type? [closed]

Let $\xi$ be a $\mathbb{R}^n$-bundle over $B$. If $B$ is paracompact and has finite covering dimension, does it follow that every $\xi$ over $B$ has finite type?
4
votes
1answer
88 views

Grassmannian, symmetric, idempotent matrices of trace $n$?

How do I see that $G_n(\mathbb{R}^m)$ is diffeomorphic to the smooth manifold consisting of all $m \times m$ symmetric, idempotent matrices of trace $n$?
0
votes
1answer
25 views

Show $\mathbb{R}^2$ with $dr^2 + \sinh^2r d \theta^2$ is isometric to the Poincare disc with $g = \frac{4(dx^2 + dy^2)}{(1-x^2-y^2)^2}$

How do I show $\mathbb{R}^2$ with $dr^2 + \sinh^2r d \theta^2$ is isometric to the Poincare disc with $g = \frac{4(dx^2 + dy^2)}{(1-x^2-y^2)^2}$? I tried converting to polar coordinates, so that $x = ...
2
votes
1answer
39 views

how to solve a pde whose coefficient is the function itself

I am studying differential geometry, Walker metric in three dimension. I try to find the geodesic equations of a Walker manifold and I need to solve the following PDE. Unfortunately, I didn't take any ...
1
vote
0answers
16 views

A question about notation in derivatives inn $\partial_{\bar{A}}L^I$

In this paper http://arxiv.org/abs/1210.2332, when the authors say eq (2.17): $$\partial_{\mu}L^I=\partial_{\bar{A}}L^I\partial_{\mu}\bar{z}^A+\partial_AL\partial_{\mu}z^A$$, does they mean by ...
2
votes
1answer
26 views

How can I prove that for a Killing vector $\nabla^a \nabla_a \xi^\mu = -R^b_a \xi^a$?

I'm taking a course on General Relativity and I'm trying to prove that for a Killing vector field $\xi^\mu$ the following equation holds: $$\nabla^a \nabla_a \xi^\mu = -R^\mu_a \xi^a$$ Where ...
1
vote
1answer
34 views

Compactification of Lie Group

Is there a way to embed a Lie Group $G$ into a compact lie Group $H$, such that the inclusion is a Lie group homomorphism?
5
votes
1answer
36 views

Can any smooth function be written in this form?

Can any smooth function $F: \mathbb{R}^n \to \mathbb{R}$ be written in the form$$F(x) = F(a) + \sum_{\mu = 1}^n (x^\mu - a^\mu)H_\mu(x),$$where $a = (a^1, \dots, a^n) \in \mathbb{R}^n$ and the $H_\mu$ ...
2
votes
0answers
22 views

Connection on $T\mathbb{R}^n$

Let $\nabla$ be a connection on the tangent bundle $T\mathbb{R}^n$. Now, I need to show that there exist smooth function $C_i: \mathbb{R}^n\to \mathbb{R}^n\times \mathbb{R}^n$, $i=1,\dots ,n$ such ...
2
votes
1answer
369 views

Projected Area of Circle onto the Side of a Cylinder

I have encountered a problem that requires me to find the projected area of a beam of light (circular cross-section, with radius R1) vertically onto the side of a cylinder with R2 as its ...
1
vote
0answers
24 views

Proving smooth map between smooth manifolds is constant based on push forward being zero

I have just me this problem in my class on smooth manifolds from Lee's introduction to smooth manifolds, from the chapter on the tangent bundle stating the following: Let M, N be smooth manifolds, ...
1
vote
1answer
26 views

Show $F(x,y,z) = (x^2-y^2,xy,xz,yz)$ gives an embedding of $\mathbb{R}P^2$ into $\mathbb{R}^4$

Define $F: \mathbb{R}^3 \rightarrow \mathbb{R}^4$ by $F(x,y,z) = (x^2-y^2,xy,xz,yz)$. Notice that F(x,y,z) = F(-x,-y,-z) so that $F|_{S^2}$ defines a mapping $\tilde{F}: \mathbb{R}P^2 \rightarrow ...
-2
votes
1answer
38 views

A function $\phi$ between two manifolds of class $C^\infty$ is constant if $d\phi$ [closed]

Let $M$ and $N$ two manifolds of $C^{\infty}$, $M$ connected, and $\phi:M\to N$ also of class $C^{\infty}$ so that in all point $m\in M$ the function $d\phi(m):M_m\to N_{\phi(m)}$ between the ...
2
votes
1answer
50 views

confusion on exercises from LEE's Book on Riemannian Manifold

I am reading John Lee's "Riemannian Manifolds an Introduction to Curvature" . At page 15 exercise 2.3 asks to prove that there exist a smooth extension of a function defined on a embedded ...
1
vote
2answers
35 views

Riemannian metrics on homogeneous spaces

Let G be a Lie group and H be a compact subgroup. The (left) coset space G/H is, up to an isomorphism, equivalent to the smooth homogeneous manifold M. My question is, is it possible to impose an ...
1
vote
1answer
27 views

Point of intersection of tangents

If the distance of two points $P$ and $Q$ from the focus of of a parabola $y^2 =4ax$ are $4$ & $9$ then what is the distance of the point of intersection of tangents at $P$ and $Q$ from the ...
0
votes
1answer
43 views

Local diffeomorpism is a covering?

$D:\tilde{M}\rightarrow \Omega$ is a local isometry (developing map) onto a connected manifold $\Omega$. $\tilde{M}$ is simply connected, has constant sectional curvature and there exists a deck group ...
2
votes
0answers
38 views

Classify the set $\{(x,y,z):f(x,y,z)=0, \nabla f=0\}$, where $f$ is a polynomial of degree at most 3.

Suppose that $f(x,y)$ is a nonzero polynomial of degree at most 2. Observe the following set: $$S=\{(x,y) : f(x,y)=0 ,\; \partial_{x}f(x,y)=0,\; \partial_{y}f(x,y)=0 \}.$$ Note that this set is the ...
1
vote
1answer
48 views

Stokes theorem for Cuboid

I need to proof stokes theorem $\int_Qd\omega=\int_{\partial Q}\omega\;$ for a 2-form and $Q\subset \mathbb R^3 \;$a cuboid. Since $\omega \;$ is a two form it can be written as $$\omega ...
0
votes
1answer
35 views

Differential of a smooth function on a manifold

Let $S^2$ be the sphere in $\mathbb{R}^3$, let's consider the (inverse) chart $\varphi$ $$x=\sin v\cos u, y=\sin v \sin u, z=\cos v$$ now let $f$ be the restriction of the linear aplication of ...
0
votes
2answers
23 views

Expressing tangent curve via level surface and graph of function

Given the sphere of radius $2$ centered at $(2,-1,0)$, find an equation for the plane tangent to it at the point $(1,0,\sqrt{2})$ in the following ways: 1) by considering the sphere as the graph of ...
1
vote
1answer
46 views

Differential of a form

I'm learning about Chern-Simons theory and my differential geometry is a bit rusty. The Chern-Simons 3-form is given by $\omega_3=tr(A\bigwedge\nolimits dA+\frac{2}{3}A\bigwedge\nolimits A ...
0
votes
1answer
29 views

Differential Geometry Proof Regarding Arclength, Tangents, Curvature, and Parameters

Consider a regular curve q(t) with arclength parameter s. Show that if $T(t_{n}) \neq T(t_{0})$ and $t_{n} \rightarrow t_{0}$, then $$1 = lim_{t_{n} \rightarrow t_{0}} \frac{|\theta(t_{n}) - ...
4
votes
2answers
62 views

If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, how do I see that $n$ must be of the form $2^r - 1$ or $2^r - 2$?

If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, how do I see that $n$ must be of the form $2^r - 1$ or $2^r - 2$? For starters, I know that if the $n$-dimensional $M$ can be immersed in ...