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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1answer
15 views
Show that an Ehresmann connection on a principal G bundle is equivalent to a Lie Algebra Valued one form.
Let $E$ be a smooth principal $G$-bundle on M.
The vertical bundle $V$ is defined as $V=\ker(d\pi:TE\to \pi^*TM)$. An Ehresmann connection on $E$ is a smooth subbundle $H$ of $TE$ (also called the ...
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0answers
15 views
uniformization theorem - squares and circles
I am trying to understand the uniformization theorem and get some intuition about it.
The uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of ...
3
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0answers
31 views
The derivative of a family of flows
If one has a family of flows, can one describe the derivative in the "family" direction?
Specifically, let $M$ be a smooth manifold and let
$X_{s,t}$
be a 2-parameter family of fields on $M$. That ...
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0answers
46 views
Operations on vector spaces
Is there a symmetrical analogue of the grassmann tensor products? Is it the polyadic tensor product? Are such symmetrical products used anywhere in differential geometry?
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1answer
56 views
Prove that the tangent space has the same dimension as the manifold
I asked this question a couple of days ago. And I thought that I totally understood the question. However it turned out that I didn't, since the argument I constructed was proved to be wrong just now: ...
1
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3answers
72 views
Diffeomorphism preserves dimension
I read from Milnor's book $\textit{Topology from the Differentiable Viewpoint}$ this assertion
"If $f$ is a diffeomorphism between opensets $U\subset R^k$ and $V\subset R^l$, then k must equal l, and ...
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0answers
41 views
Definitions of Semisimple Lie Algebra
We usually define semisimplicity of a Lie algebra $\mathfrak{g}\subset M_n ({\bf R})$
from two ways. I want to know the relation between them.
One of the definitions of semisimple Lie algebra is ...
4
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2answers
69 views
How do I know when a form represents an integral cohomology class?
Suppose $M$ is an $n$-dimensional manifold, and $\omega \in \Omega^p(M)$ is a closed $p$-form. Moreover, assume that $d\omega = 0$, so that it represents a de Rham cohomology class.
I would like to ...
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1answer
25 views
Relating the curvature of a plane curve to the curvature of a stretched version
Let $\theta : I \to \mathbb{R}^2$ be a regular plane curve with curvature $ |k_{\theta}|\leq1$ everywhere.
We now define a curve $\theta_{d}$ by stretching $\theta$ in one direction, i.e.,
$\theta = ...
3
votes
1answer
45 views
Equivalence of Definitions of Principal $G$-bundle
I've finally gotten around to learning about principal $G$-bundles.
In the literature, I've encountered (more than) four different definitions. Since I'm still a beginner, it's unclear to me whether ...
1
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1answer
44 views
Pullback Calculation
If we define the 2-form $\omega=\frac{1}{r^3}(x_1dx_2\wedge dx_3+x_2dx_3\wedge dx_1+x_3dx_1\wedge dx_2)$ with $r=\sqrt{x_1^2+x_2^2+x_3^2}$
If we now define ...
1
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0answers
28 views
Nilpotent infinitesimals comparison
I'd like to understand better the advantages and disadvantages of various approaches to nilpotent infinitesimal numbers and their application to differential geometry in the context of physics and ...
2
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0answers
27 views
Uniformly quasi-isometric patches over classes of Riemannian manifolds
Suppose $(M^d,g)$ is a closed, connected Riemannian manifold.
Is there a constant $R > 0$ such that for all $z \in (M,g)$, for all $x \in B_R(z)$, \begin{equation} \frac{1}{2} \lVert \xi ...
1
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1answer
65 views
Prove that the dimension of the tangent space $T_x(X)$ of a k-dimensional manifold is k
In section 2, page 9 of Guillemin and Pollack's book $\textit{Differential Topology}$, he gave a proof that the dimension of the tangent space $T_x(X)$ is equal to the dimension of the manifold $X$. ...
0
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1answer
67 views
Second Fundamental Form of Torus
I want to prove that the mean curvature vector of the flat Torus $T^n=S^1\times S^1\times ...\times S^1\subset\mathbb{R}^{2n}$ is zero. But my calculations show me to the contrary that the mean ...
1
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1answer
39 views
Compactness of semisimple Lie algebra
I want to prove that on a semisimple Lie algebra $\mathfrak{g}$ over ${\bf R}$:
$\mathfrak{g}$ is compact if and only if the Killing form is strictly
negative definite.
Here the Lie algebra is ...
13
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4answers
152 views
How to show in a clean way that $z^4 + (x^2 + y^2 - 1)(2x^2 + 3y^2-1) = 0$ is a torus?
How to show in a clean way that the zero-locus of $$z^4 + (x^2 + y^2 - 1)(2x^2 + 3y^2-1) = 0$$ is a torus?
2
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1answer
28 views
Show a certain vector field $V$ only commutes with others that are collinear with $V$
I found this on an old qualifying exam. I started the problem, but I'm not sure what my next step should be:
Let $T^2$ be the standard 2-dimensional torus with $\mathbb{Z}$-periodic coordinates ...
0
votes
0answers
18 views
Prolate spheroidal coordinates
If $\alpha \in (0,\infty)$, $\beta \in (0,\pi)$ and $\theta \in (0,2\pi)$.
$$\varphi (\alpha,\beta,\theta) =(
\sinh(\alpha)\sin(\beta)\cos(\theta),
\sinh(\alpha)\sin(\beta)\sin(\theta),
...
1
vote
1answer
37 views
Diffeomorphic surfaces and Jacobian
Suppose $S$ and $T$ are bounded (open) surfaces in $\mathbb{R}^n.$ Let them have boundary $\partial S$ and $\partial T$.
Suppose $F:S \to T$ is a $C^k$ diffeomorphism.
Under what conditions on $F$ ...
1
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0answers
22 views
Friedrichs extension of Schrödinger operator
the Dirichlet-Laplacian on a compact connected Riemannian manifold M with boundary $\partial M$ is defined via the Friedrichs extension of the Laplacian $-\Delta: C^{\infty}_0(M)\subset L^2(M) ...
2
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0answers
75 views
Group actions and associated bundles
Let $P$ be a principal $G$-bundle over $B$, and let $G$ act on some space $F$ (feel free to work in your favorite category of spaces, if this helps). Then $\text{Aut}{P}$ (aka the group of gauge ...
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1answer
24 views
Derivations on semisimple Lie algebra
First recall some definitions :
Let $B$ be a Killing form on Lie algebra $\mathfrak{g}$ over ${\bf R}$ such that
$B(X,Y)\doteq Tr(ad_Xad_Y)$.
$\mathfrak{g}$ is semisimple if $B$ is ...
1
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2answers
36 views
Vector Field Generating Variation Along Curve
I'm learning a proof of the fact that length extremising curves are geodesics of the Levi-Civita connection, and have found something I don't understand. The argument states the following.
Suppose ...
3
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1answer
46 views
Question about diffeomorphism
Here is an assignment problem:
$f:\mathbb{S}^2 \longrightarrow \mathbb{S}^2$ is smooth and surjective. Prove $\exists$ open subset $ U $ of $\mathbb{S}^2$, such that $f|_U$ is a diffeomorphism.
I've ...
4
votes
1answer
83 views
How to understand $\frac{d}{dt}\{(\exp(tX))_*(Y)\}|_{t=0}=[X,Y]$?
Let $G$ be a Lie group on which $X$ and $Y$ are two vector fields. Let $G\xrightarrow{\exp(tX)} G$ be the (Lie theory) exponential map corresponding to $X$. Then of fundamental importance is ...
1
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0answers
72 views
Compactness property
Let $\Omega \subset X$, X: Banach space. Given $\varepsilon \ge 0$, we define the set of $\varepsilon-normals$ to $\Omega$ at $\bar{x}$$\in \Omega$ by:$\widehat N_\varepsilon(\bar ...
4
votes
0answers
55 views
Differentiable manifolds, Serge Lang
I have started reading "Introduction to differentiable manifolds" by Serge Lang. In this book, Lang takes a different approach, by immediately introducing manifolds on arbitrary Banach spaces. His ...
6
votes
0answers
71 views
Invariant submanifolds
Let $M$ be a smooth manifold, and let $N$ be a submanifold. Let $V$ be a smooth vector field on $M$ which generates a flow $\Phi_t$ on $M$. My intuition tells me (perhaps modulo some technical ...
2
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0answers
21 views
Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary?
Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary? What smoothness is required of the boundary? I would be grateful for some references to this.
2
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2answers
51 views
Smooth maps on a manifold lie group
$$
\operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\
\begin{align}
&n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\
&n = 2, \operatorname{GL}_n(\mathbb ...
3
votes
0answers
44 views
Alternative rigorous definition of a surface integral
Consider some open subset $U$ of $\mathbb{R}^n$ where $U$ has a (piecewise) $C^1$-boundary. Let $f$ be some smooth (enough) real function. Is there some way to give a measure-theoretic definition of ...
4
votes
1answer
33 views
Complexified tangent vector, complex manifold
Consider a complex submanifold $M$ of a complex ambient vector space $X$. Suppose you have a base point $p \in M$ and a $C^1$ arc $\gamma(t)$ passing through $p$ and staying in $M$, with tangent ...
6
votes
1answer
51 views
Hamiltonian for Geodesic Flow
I'm trying to prove that geodesic flow on the cotangent bundle $T^* M$ is generated by the Hamiltonian vector field $X_H$ where
$$H = \frac{1}{2}g^{ij}p_i p_j$$
but I am stuck. Could somebody show ...
1
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1answer
32 views
Space of embedded surfaces with a common point
Consider the space of all embedded orientable surfaces in $ R^3 $ of constant mean curvature (the minimal case is included) passing through the origin. I'm asking if there exists a topology on this ...
1
vote
2answers
39 views
What does generic immersion mean?
I have been looking for the meaning of generic immersion
In the textbook I am reading, a theorem involves a curve with y coordinate satisfying $y'(0)=0$ says at some stage the following: "...Since ...
2
votes
2answers
55 views
Symplectic Form Preserved by Orthogonal Transformation
I'm trying to prove that the symplectic form
$$\omega = d(\cos\theta) \wedge d\phi$$
is preserved by the action of $SO(3)$ on $S^2$ where $\phi$ and $\theta$ are spherical polars. Now $SO(3)$ simply ...
2
votes
0answers
32 views
No flux boundary condition on PDE on surface (Laplace-Beltrami)
What would a Neumann BC on a PDE posed on a surface look like? In the flat case, we have
$\nabla u \cdot N = 0$
where $u$ is the solution of the PDE and $N$ is unit normal vector.
In a surface case, ...
1
vote
1answer
42 views
What is a conormal vector to a domain intuitively?
I read that a conormal vector of a domain is a vector that is tangential to the domain and normal to its boundary.
If we consider an open disk in $\mathbb{R}^2$ what is a conormal vector at a point ...
1
vote
1answer
52 views
Every curve is a geodesic??
I've been reading up on how isometries send geodesics to geodesics. I recently saw a proof of another theorem that used the fact:
The set of fixed points of an isometry is a geodesic.
But isnt the ...
0
votes
1answer
12 views
Inequality in proving the Isoperimetric Inequality
The question came from reading the following post: A proof of the Isoperimetric Inequality - how does it work?
I almost can follow the whole proof, but I am stuck at one point. Why does $(x^2 ...
1
vote
1answer
24 views
How to directly show that Figure 8 injective immersion is not a monomorphism
I'm working in the category of smooth manifolds. The injective immersion that takes the open unit interval $(0,1)$ to the figure 8 is a well-known example of an injective immersion that is not an ...
-1
votes
2answers
99 views
The euclidean space $\Bbb R^n$ is orientable as a manifold.
I know that
The euclidean space $\Bbb R^n$ is orientable as a manifold.
I think that it is orientable because it has a nowhere vanishing $n$-form.
But I am not sure.
Please can you explain ...
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0answers
28 views
maxima of the sum of unimodal functions .
I have a set of unimodal functions. Each function has real roots. All roots of each function lie outside a certain limit points. These limit points are the same for each function. Each function is in ...
5
votes
0answers
46 views
What is the volume of Complex Projective Space with Fubini-Study Metric?
I try to compute the volume of the complex projective space $\mathbb{CP}^n$ with Fubini-Study metric, normalized to have diameter $=\pi/2$ i.e. the sectional curvatures lie between $1$ and $4$. Fix a ...
5
votes
1answer
56 views
Applications of information geometry to the natural sciences
I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has ...
2
votes
1answer
68 views
Differential Geometry Video Lectures
I know there's a similar question here, however since what I found there wasn't what I was looking for I thought on creating a new question. I'm studying Differential Geometry through Spivak's book "A ...
1
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0answers
27 views
Following a polyline along the surface of a polygon that is twisted
I have (hopefully) an interesting problem regarding geometry. I will also search online and in literature but I thought to pose the question here as a third resource.
For my problem I need to get the ...
2
votes
0answers
34 views
Stoke's theorem application to curl theorem. I did. Please can you check it?
Now, I need to apply stoke's theorem to curl theorem.
My teacher gave a hint.
Accourding to the hint, I accept $w=Pdy∧dz +Q dz∧dx + R dx∧dy$ $\in Ω^2(M)$
$dim(M)=2$
M is the subset of $\Bbb ...
2
votes
0answers
51 views
Real projective space is Hausdorff
I could not understand the proof of thıs proposition can you help me and give clear explanation.Just can you say how we have (n+1)x2 matrix??
This prove is correct or I need to add something ?? ...
