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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Is Floer homology always isomorphic to the singular homology of some space?

After I studied Morse homology, I'm now studying the following Floer homology theories : 1) Symplectic Floer homology ; 2) Floer homology of lagrangians ; 3) Heegard-Floer homology ; ...
2
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1answer
44 views

A book on Vector Calculus with emphasis on geometrical intuition

I am a physicist trying to learn vector calculus in a way that is a mixture of the way mathematicians learn it with the way that physicist learn it in order to be able to learn Differential Geometry ...
3
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2answers
39 views

convention of a default atlas

Recently, I have been studying the basics of differential geometry and te necessary preliminaries. I arrived at the construction of differential structure on topological manifolds, where the non-...
1
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1answer
32 views

smooth manifolds, equivalent statements

Let $X,Y$ be smooth manifolds. Show: A function $f:X\to Y$ is smooth, iff for every open $V\subseteq Y$ and every smooth function $g:V\to\mathbb{R}$ the composition $g\circ f: f^{-1}(V)\to\mathbb{R}$ ...
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1answer
27 views

covering space, smooth manifold

Let $p:Y\to X$ be a covering space and $p^{-1}(x)$ countable for every $x\in X$. Task: Let $X$ be a smooth manifold. Show, that $Y$ has the structure of a smooth manifold, regarding this $p$ is ...
1
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1answer
31 views

Rotaion Surfaces and Complex Numbers

Consider a continuous invertible map $\varphi:\mathbb{R}^+ \longrightarrow \mathbb{R}$, and define the follwing surface $$ s:\mathbb{C} \longrightarrow \mathbb{R} \times \mathbb{C} $$ $$ \qquad xe^{i\...
7
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2answers
207 views

Is this contraction of metric tensor derivatives symmetric?

A couple of times when I've tried to prove symmetries of various tensors (for learning), I've ended up with the expression below, and the fact that either a) I made mistake, or b) the expression is ...
2
votes
1answer
38 views

Incompatible first and second fundamental forms

Say the first and second fundamental forms of a surface (a and b) in 2D are incompatible (i.e. they do not satisfy the Codazzi-Mainardi equations), then the "surface" cannot be embedded in 3D. Is this ...
5
votes
1answer
66 views

Is the contraction of an harmonic form harmonic?

As I'm still a beginner in complex differential geometry, as soon as I tried reading an article I got stuck on what (I think) should be a minor detail. I hope someone can help me a bit. Let $M$ be a ...
4
votes
1answer
104 views

Diffeomorphisms between factors in diffeomorphic product manifolds

Let $M$, $N$ and $P$ be three smooth manifolds such that $M \times N$ is diffeomorphic to $M \times P$. I need to know about some conditions under which one can deduce that $N$ is diffeomorphic to $P$....
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vote
1answer
41 views

Smooth structure on open subsets of manifolds

Let $X$ be a smooth manifold and $U\subseteq X$ open. Define a canonical smooth structure on $U$, for which the embedding $U\to X$ is smooth. Hello, I want to solve this task. My try was as follows:...
3
votes
1answer
63 views

Does the associated bundle functor have left or right adjoints?

Let $\mathsf{Prin}_G$ be the category of (right) $G$-principal bundles, with a morphism from the bundle $p: P \to M$ to the bundle $p': P' \to M'$ being a pair of arrows $\chi: P \to P'$ and $\bar{\...
3
votes
1answer
61 views

Automorphisms action on $\mathbb C$

If we know that the $$Aut \ \mathbb C=\{z\mapsto az+b:a\in \mathbb C^*,b\in \mathbb C\}$$ What are the subgroups of $Aut \ \mathbb C$ that act withiut fixed points and properly discontinuously on $\...
1
vote
1answer
28 views

The action of automorphisms on the Riemann sphere

If we are given that the automorphism group of the Riemann sphere is $$Aut\ \mathbb P^1=\{z\mapsto \frac{az+b}{cz+d}:ad-bc=1\}$$ Why this group does not have any proper subgroups that act without ...
3
votes
2answers
74 views

Defining a Riemannian metric

Let $M$ be a smooth manifold. I have seen a Riemannian metric be defined in many ways: A smooth choice of an inner product $g_p:T_pM\times T_pM\to\mathbb{R}$ which is symmetric and positive-definite,...
-4
votes
2answers
38 views

Is $T_pM$ isomorphic to $T_{F(p)}N$?

Let $M$ and $N$ be two smooth n-dimensional manifolds and $F:M\to N$ be a diffeomorphism. Is it true that $F_{*p}:T_pM\longrightarrow T_{F(p)}N$ is an isomorphism?
3
votes
1answer
34 views

Ratio of holomorphic forms on a Riemann surface

Let $R$ be a Riemann surface of genus $g>1$ and $\omega$, $\sigma$ two holomorphic 1-forms on $R$. This means that locally we can write $\omega=fdz$ and $\sigma=gdz$ with $f$ and $g$ holomorphic. ...
0
votes
0answers
21 views

Coordinates of exterior derivative of dual basis of local frame for the tangent bundle

Let $M$ be an $n$-manifold. Let $E_1, E_2,\dots, E_n : U\subset M \to TM $ be a local frame for $TM$ with associated local dual frame $\epsilon^1, \epsilon^2,\dots, \epsilon^n : U\subset M \to T^*M $. ...
1
vote
1answer
18 views

extending functions from the horizontal bundle to the whole bundle

Let $(M,g)$ be a Riemannian manifold and $G$ a compact Lie group acting freely and isometrically on $M$. Let $\pi \colon M \to M/G$ be the projection to the orbits. Using the metric, we get a ...
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0answers
39 views

Quick question: Curvature form of a connection on the trivial bundle

Let $L=\mathbb{R}^2\times U(1)$ be the trivial $U(1)$-bundle over $\mathbb{R}^2$. Define a connection $\nabla=d+A$ where $A=fdx+gdy$ is an $\mathbb{R}$ valued $1$-form on $L$. That is, $\nabla$ gives ...
1
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1answer
66 views

How to build smooth variations along arbitrary paths at the endpoint?

Let $M$ be a smooth manifold, $\gamma:[0,a] \to M$ a smooth path. Assume we are given another smooth path $\phi:(-\epsilon,\epsilon) \to M$ that starts from an endpoint of $\gamma$, i.e $\phi(0)=\...
2
votes
2answers
55 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 ...
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0answers
28 views

Existence of submersions between manifolds

I have a ton of problems, where I need to prove (or disprove) the existence of submersions between given manifolds. I will give you some examples, and hopefully I can learn the techniques to solve all ...
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0answers
23 views

Existence of map between projective planes.

I'm struggling to solve this problem, any indications would be appreciated. Is there an application $f : \mathbb RP^3 \to \mathbb RP^1$ of class $\mathcal C^3$ such that $f^{-1}(p)$ is the union of ...
0
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0answers
26 views

Derivative of projection

Let $\Omega \subset \mathbb{R}^n$ be a limited domain of class $C^\infty$ (open, connected) and $\varepsilon > 0$ small such that $$ \Omega_\varepsilon = \{y \in \overline{\Omega} : d(y, \partial \...
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0answers
24 views

Roadmap to Differential Geometry for Machine Learning

Recently within machine learning, there are a lot of works on non-convex optimization and natural gradients methods etc which are based on differential geometry, it gives rise to increased need to ...
0
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1answer
34 views

Definition of formal adjoint of covariant derivative

I read in Einstein Manifolds, L. Besse that the covariant derivative $D: \mathcal{J}^{(r,s)}(M)\to \Omega^1(M)\otimes\mathcal{J}^{(r,s)}(M)$ admit an formal adjoint $D^*:\Omega^1(M)\otimes\mathcal{J}^{...
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0answers
26 views

Clarification for Manifold and Manifold with Boundary Definitions

I am reading Spivak's Calculus on Manifolds, and just need a bit of clarification when it comes to definitions. He defines a manifold as some space $M$ that satisfies: $(M)$: $\forall x \in M, \...
0
votes
1answer
30 views

Compute Christoffel symbols of a rotating cartesian coordinate system

Suppose we have a smooth manifold $(\mathbb{R}^3, \mathcal{O}_{\mathbb{R}^4}, \{(\mathbb{R}^3,x),(\mathbb{R}^3,y)\},\nabla,t)$ where $t:\mathbb{R}^3\rightarrow\mathbb{R}$ is such that $t(a,b,c,d)=a$, $...
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0answers
17 views

Unique Solution to Equation in Two Variables & Possible Use of the Implicit Function Theorem

Let $g(x) : R \to R$ be a continuous function; Consider the equation $ T(x,y) = y^3 -y^2 +(1+x^2)y - g(x)$ Show that for a given $x$ there exist a unique solution $y$ to the equation $T(x,y) = 0$. ...
0
votes
1answer
36 views

Why is the Christoffel symbol of the 2nd kind symmetric in lower indices?

I have consulted multiple books on tensors for physicists, but they all take for granted this relation: $\Gamma_{ij}^k = \Gamma_{ji}^k$ However, no proof is provided and I cannot find a single one ...
1
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1answer
19 views

Given parametrization of torus is equal to level surface

I need to show that the torus $T=\pmatrix{(R+r\cos\phi)\cos \theta\\(R+r\cos\phi)\sin\theta\\r\sin\phi}$ is equal to the surface given implicitly by $(\sqrt{x^2+y^2}-R)^2+z^2-r^2=0$. I already got ...
3
votes
1answer
91 views

Need help understanding local coordinates of differential forms

I was trying to check a variation of theorem 2.2: I did the first part of the proof using the standard contact structure on $\mathbb R^{2n + 1}$ which worked out as expected. Then I wanted to do it ...
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0answers
39 views

Pullback Courant algebroid

I am reading the lecture notes on generalized geometry http://www.staff.science.uu.nl/~caval101/homepage/Research_files/australia.pdf but now I am stuck on the exercise 1.31 at page 13. Here is the ...
5
votes
1answer
49 views

Is a surjective mapping of R2 to itself with full rank derivative everywhere necessarily injective?

If $f:\mathbb R^2\rightarrow\mathbb R^2$ has rank 2 derivative everywhere, then by the inverse function theorem it is locally injective. If it is surjective, is it then necessarily globally injective ...
1
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1answer
131 views

Envelope of family of curves $x(u,v)=\cos^2(u)\cos(v)+\cos(u)\sin(u)\sin(v)$, $y(u,v)=\cos^2(u)\sin(v)-\cos (u)\sin(u)\cos(v)$

How to generally find singular solution or envelope of a two parameter family of curves $ x(u,v),y(u,v) $ in the plane? The parametric equations $$x(u,v) = \cos^2 (u) \cos (v) + \cos (u) \sin (u) \...
1
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0answers
29 views

Smooth structure in reconstruction theorem

Let $M,F$ be smooth manifolds, $\{U_i:i\in I\}$ an open cover of $M$ and a cocycle $\{t_{ij}:U_i\cap U_j\to\mathrm{Diff}(F)\}$. In almost any book which discusses fibre bundles, one can find the ...
0
votes
1answer
26 views

Curve in a disk

I take a curve $\vec\gamma:[a,b]\longrightarrow \Delta$, where $\Delta \subseteq \mathbb{R}^2$ is the disk of radius $r>0$. If the curve has length $L>0$ does exist an upper bound (in terms of $...
1
vote
0answers
28 views

Volume of a geodesic ball in a Riemannian manifold with $K<0$.

Let M be a simple connected Riemannian Manifold wih $K_M < 0$. Prove that the volume of any geodesic ball of M is strictly greater than $\frac{Vol(S^{n-1})r^n}{n}$, where $n = dim(M)$ and $r$ is ...
1
vote
1answer
30 views

Do these assumptions on a mapping ensure it is a diffeomorphism?

$A\subset \mathbb{R}^m$ is an open and bounded set, $f:A\longrightarrow \mathbb{R}^n$, $m\leq n$, is injective, continuously differentiable and its Jacobian matrix has full rank on $A$. Does this ...
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0answers
77 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 $\gamma'_1(1)...
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2answers
29 views

curl-free vector field on 3-torus

Let $U$ be an open and simply-connected subset of $ \mathbb{R}^3$. Then for every curl-free vector field $v \: \colon U \to \mathbb{R}^3$ there is a potential $\phi \in C^{\infty}(U; \mathbb{R})$ such ...
5
votes
3answers
1k views

divergence of a vector field on a manifold

I've been asked to show the following: For a vector field $V$ on a semi-Riemannian manifold with metric $g$ that $$Div \cdot V = \frac{1}{\sqrt{\det(g)}}\partial_i\left(\sqrt{\det(g)}V^i\right)$$ I ...
1
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1answer
29 views

Geodesic sphere in $\mathbb H^2$

I saw the definition of a geodesic sphere, and I think I'm not able to "see" how do they look like. For example, it's obvious that in $\mathbb R^n$ geodesic spheres are simply normal spheres, and that ...
3
votes
2answers
158 views

Reference of what metric can be placed on manifold?

I just read some conclusion that $T^2$ can't be placed metric with positive curvature at all points. I don't know why is so . And what book introduce about this ? I mean about what metric can be ...
0
votes
2answers
400 views

How to prove a map is a local diffeomorphism

Let $F:U \subset R^2 \rightarrow R^3$ by given by F(u,v) = $(u\sin{(\alpha)}\cos{(v)},u\sin{(\alpha)}\sin{(v)},u\cos{(\alpha}))$, (u,v) $\in U$ = {(u,v) $\in R^2 \space; u > 0$} with $\alpha$ a ...
0
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1answer
14 views

Existence of a $G$-invariant metric on a principal bundle

Given a smooth principal $G$-bundle $\pi: P \rightarrow M$, I want to show the existence of connections by showing that $P$ admits $G$-invariant metrics. I was thinking in a kind of averaging ...
0
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1answer
30 views

How to prove that a space is not a differential manifold?

Given a box (the surface of a cubic) in R^3 space, can I give a smooth structure on it to make it a differential manifold?
2
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0answers
53 views

Is this what universal covering spaces are used for?

From the perspective of real analysis, we have: $$\int_{-1}^{1}\frac{1}{1+x^2} = \mathrm{tan}^{-1}(1)-\mathrm{tan}^{-1}(-1) = 2\mathrm{tan}^{-1}(1) = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2}$$ Something ...
2
votes
1answer
48 views

Partial Derivative of Line Integral as a Potential of F

Context to the question: Say $ \{F_{k} \} \to F$ uniformly on a compact subset $K \subset T$, for $ \{F_{k} \}$ a sequence of conservative vector fields and $T$ open and connected. I've shown that ...