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)

1
vote
0answers
36 views

Invariant connection form

Let $\pi : E \to \mathbb{R}^3$ define a vector bundle with a connection form $\nabla$ on $\mathbb{R}^3$. The text that i'm am reading then goes on to say that $\nabla$ is $\mathbb{R}$-invariant in ...
1
vote
1answer
38 views

Determining when a differential form is closed

I'm looking at the $3$-form on $\Bbb R^4 \setminus \{0\}$ defined by $$ \gamma_k = \frac{1}{\Vert x \Vert^{2k}} i_E(dV),$$ where $k \in \Bbb R$, $E$ is the Euler vector field $x^i \frac{\partial}{\...
2
votes
0answers
25 views

Trajectories of a vector field on the 2-sphere

Consider the vector field given by given by $(-zx,zy,0)$, where we've identified $T_pS^2$ where we've identified the space of vectors orthogonal to $p$. How do we visualize the trajectories of the ...
1
vote
0answers
26 views

Product of currents

De Rham currents are for differential forms as distributions are to (smooth) functions. There is a notion of exterior product for differential forms: I wonder whether there is an algebra structure on ...
7
votes
1answer
151 views

Green's Function for Laplacian on $S^1 \times S^2$

As indicated by the title, I am looking to find the Green's function for the Laplacian on $S^1 \times S^2$. Is such a function known? If not, does anyone have an approach to constructing such a ...
3
votes
1answer
67 views

Vector bundles and de Rham cohomology

So $M$ is a compact manifold and I am asked to either prove the following statement or give a counterexample: if $\pi: E \rightarrow M$ is a vector bundle, then $H^2(E) \simeq H^2(M)$. I know the ...
0
votes
1answer
30 views

Differentiable functions between manifolds are continuous

Let $f:M \to N$ be differentiable function between manifolds. I want to show that $f$ is continuous. First, that $f$ is continuous should mean (correct me if I'm wrong!) that for every point $a\in N$ ...
3
votes
1answer
35 views

Sections of tensor bundle are tensor product of sections

Given $E,F$ vector bundles over a manifold $M$, I would like to know a proof of $\Gamma(E\otimes F) = \Gamma(E) \otimes_{C^\infty(M)} \Gamma(F)$. Where $\Gamma$ denotes the smooth sections over $M$. ...
1
vote
0answers
25 views

Is this the correct way to compute tensor bundles of smooth manifolds given by a smooth function?

Let $M$ be a smooth submanifold of $\mathbb{R}^n$ given by the vanishing locus of a smooth function $f(x_1,\ldots,x_n)$. I can compute the cotangent bundle from this embedding by looking at the ...
0
votes
1answer
39 views

Construction of connections given in “From Calculus to Cohomology”

I'm struggling to understand part of the descripition of a connection on page 167 of Madsen, Tornehave "From Calculus to Cohomology". Immediately after defining a connection $\nabla$ on a bundle $\xi$ ...
0
votes
0answers
15 views

Velocity of a 2-parameter curve

Let $M$ be a manifold and $I,J$ be two intervals on $\mathbb{R}$. Suppose $\alpha:I\times J\longrightarrow M$ a smooth map. It is clear that $s\mapsto\alpha(s,t_0)$ and $t\mapsto \alpha(s_0,t)$ are ...
1
vote
1answer
29 views

The dimension of a projectivised, complexified tangent bundle

I am looking at page 15 here, starting from the line 'Let us suppose we are given a...' The Zoll projective structure is not relevant to this question, so don't worry about that. We take, in this ...
4
votes
2answers
106 views

Why are Lie groups automatically analytic manifolds?

In the book by Kolar, Michor, and Slovak, it is shown that multiplication $\mu:G\times G\to G$ is analytic in some neighborhood of $e$. Specifically, they show that in the chart given by $\exp^{-1}$, ...
3
votes
2answers
56 views

Elementary question: local computation of curvature on principal bundle

Let $G$ be a Lie group and $S=[0,1]^2$. Let $\omega$ be a connection $1$-form on the trivial principal $G-$bundle $P=S\times G$ over $S$. Let $(x_1,x_2)$ be coordinates on the base $S$. We can choose ...
12
votes
0answers
208 views

Derivation of a representation through a vector field

Question: (Exercise 3.4.12 - Sharpe) Let $H$ be a Lie group, $V$ a vector space, and $\rho: H \to Gl(V)$ a representation. Let $U$ be a manifold, $X$ a vector field on $U$, and $h: U \to H$ and $f:...
0
votes
0answers
26 views

Help with derive geodesic equation

Let $(M,g)$ be a pseudo-riemannian manifold and $p,q\in M$. Suppose $\alpha:[a,b]\to M$ a smooth curve on $M$ such that $\alpha(a)=p$ and $\alpha(b)=q$. If we consider: $$L[H(s,\cdot)]=\int_a^b \sqrt{...
1
vote
1answer
61 views

Maurer Cartan Form of the Heisenberg group

I'm trying to understand meaning and application of the Maurer Cartan Form, but I'm still not quite there. I'm then trying to do some examples and trying understand how it works. I begun with the ...
1
vote
1answer
41 views

Constructing symplectic structure on $T^*M$

I read the picture below, but I don't know how to get the equation above red line. Whether by using $T_{\xi_x}(T^*M)\cong T^*_xM$ ?But which isomorphism should be choice ? Then , how to check the ...
2
votes
0answers
48 views

Overview of Geometric analysis [closed]

Can anyone tell me what geometric analysis is about? After reading some articles I have a view that it uses PDE extensively for geometric problems. Am I right in this point? Also what kind of ...
0
votes
1answer
48 views

How can I equip the universal vector bundle over $\mathbf{G}(k,n)$ with a connection?

When constructing the grassmannians $\mathbf{G}_\mathbb{F}(k,n)$ for $\mathbb{F} = \{\mathbb{R},\mathbb{C} \}$ there is a natural vector bundle $$ \mathbf{G}_{k,n} \to \mathbf{G}_\mathbb{F}(k,n) $$ ...
3
votes
0answers
78 views

Relearning differential geometry

I will shortly describe my situation and than formulate the problem. From around year I am working under supervision of my professor on master thesis in differential geometry (mainly discussion of ...
0
votes
0answers
27 views

Condition to be a Geodesic up to reparametrization

Let $M$ be an $n$-manifold and $\nabla$ a linear connection on $M$. Let $\sigma:I\subset \mathbb{R} \to M$ be curve such that in a local coordinate system we have $ \Sigma_{ij}\ (\sigma_k'' + \...
0
votes
0answers
21 views

solution of under-constrained non-linear system with Implicit Function Theorem or Fixed Point Theorem

Suppose $ U \subset \mathbb{R}^n$ is open and $\mathbf{f}: U \rightarrow \mathbb{R}^m$ is $C^1$ with $ \mathbf{f}(\mathbf{a}) = \mathbf{0}$, and $\mathrm{rank}(D\mathbf{f}(\mathbf{a})) = m$. Show ...
1
vote
1answer
80 views

Topology of the tangent bundle

Let $M$ be a smooth manifold, and let $\pi:TM\to M$ be its tangent bundle. We define the topology of $TM$ by declaring a subset $V$ of $TM$ to be open if and only if $\psi_\phi(V\cap\pi^{-1}(U))$ is ...
1
vote
1answer
33 views

Question about Brouwer degree under uniform convergence.

I was wondering the following: Say a smooth sequence $u_k$ on a smooth manifold converges uniformly to the limit $u$. Does $u$ preserve the Brouwer degree of the $u_k$'s? I also believe this is an ...
2
votes
1answer
22 views

Formula for the outward unit normal of a perturbed domain from D.Henry's book

I am reading D.Henry's book "Perturbation of the boundary in Boundary-Value Problems of Partial Differential Equations". At p.24, in the proof of Lemma 2.3 the following assertion is made: Let $\...
0
votes
0answers
21 views

Derive geodesic equation

Let $(M,g)$ be a riemannian manifold and $(U,\psi)$ local chart on $M$. If $\alpha=(x^1,\ldots,x^n)$ is a curve on $U$ such that verify: $$\sum_{i,j=1}^n \Big(\frac{1}{2}\frac{\partial g_{ij}}{\...
0
votes
0answers
22 views

Using calculus of variations to find a curve through the end points of a family of curves?

Let $I:=[0, 1]$ and let $\alpha:I\times I\longrightarrow X$ be a family of smooth curves on a smooth manifold $X$. Let us think of $\alpha$ as a family of curves $\{\alpha_s: I\longrightarrow X\}_{s\...
1
vote
1answer
35 views

Geodesics on a twisted surface of revolution

How is Clairaut's Law modified to define geodesics in a twisted surface of revolution ( so not axi-symmetric) $$ u \cos v, u \sin v , f(u) + T \, v ,$$ where T is a twisting constant? It appears ...
0
votes
0answers
27 views

New parametrization of surface

My reference for the following theorems is Do Carmo's book on differential geometry page 182-183 I want an example of new parametrization of a regular surface($S$) at a point $p\in S$ that its the ...
3
votes
0answers
68 views

Understanding twisted differential forms

I'm trying to understand twisted differential forms. I do know that they are like regular differential forms but under coordinate transformations they pick up an extra factor of the sign of the ...
1
vote
1answer
27 views

Killing/Isometry correspondence: Domains of flows generated by vector fields

I am wondering about the correspondence between the isometry group $\mathcal{I}$ and the Lie Algebra of Killing vector fields $\mathcal{K}$ on a pseudo-Riemannian manifold $(\mathcal{M}, \mathbf{g})$. ...
3
votes
1answer
50 views

Perturbing a regular submanifold to ensure submersion.

Suppose $M$ is a regular compact submanifold with boundary of dimension $n$ in $\mathbb R^{n+1}\smallsetminus 0$, and assume that for each $v\in S^n$ the ray emanating from $v$ intersects $M$ in at ...
7
votes
1answer
145 views

Why use geometric algebra and not differential forms?

This is somewhat similar to Are Clifford algebras and differential forms equivalent frameworks for differential geometry?, but I want to restrict discussion to $\mathbb{R}^n$, not arbitrary manifolds. ...
2
votes
1answer
81 views

$\hat{e_\mu } \cdot \hat{e^\nu } \neq \delta _{\mu} ^{\nu}$? Tensor algebra question.

Let $\hat{e_{\mu }}$ and $\hat{e^{\mu }}$ be the co- and contravariant basis vectors, respectively, for an arbitrary coordinate system Is it true that sometimes, $\hat{e_\mu } \cdot \hat{e^\nu } \neq \...
0
votes
1answer
30 views

Lie derivative of vielbein along Killing vector

We know that a vector $X$ is killing if the Lie derivative of the metric along $X$ vanishes: $\mathcal{L}_X (g_{mn})=0$ We also know that the metric can be written in terms of the vielbeins: $g_{mn}...
2
votes
0answers
42 views

Generalisation of the Poincaré Lemma

Let $\Omega \subset \mathbb{R}^3$ be an open but not simply connected domain and let $v \: \colon \Omega \to \mathbb{R}^3$ be a continuously differentiable vector field. Assume that $\textrm{curl} \, ...
10
votes
1answer
97 views

Is a bijective smooth function a diffeomorphism almost everywhere?

Suppose I have $f: M \rightarrow N \in C^{\infty}$ a smooth bijection between $n$-dimensional smooth manifolds. Does it have to be a diffeomorphism except for a set of measure 0? I think the proof ...
1
vote
1answer
31 views

Property of the covariant derivative

I am learning to use the covariant derivative. In particular, I am trying to verify the expression $${\nabla}_{b}[(\nabla^a S) (\nabla_a S)] = 2 \nabla^a S \nabla_b \nabla_a S$$ for an arbitrary ...
6
votes
1answer
91 views

Differentiation under the integral sign for volumes in higher dimensions

Consider a smooth convex/compact domain $D\subset \mathbb{R}^n$ and a smooth, concave function $F:D\to \mathbb{R}$. Then we can define the function that simply takes the volume of the upper contour ...
0
votes
0answers
33 views

What does it mean for a tangent bundle to be parallel?

I am looking at page 8 of the paper here, just below definition 2.13. I have never come across a phrase such as '$T\mathfrak{C}\subset{TM}$ is parallel' - what does it mean for a tangent bundle to be ...
0
votes
1answer
24 views

Sequence of closed sets (Milnor's proof)

I've got a question about a descending sequence of closed sets. Milnor writes in his book "FROM THE DIFFERENTIABLE VIEWPOINT". In his proof of Sards Theorem he wrote: Let $f:U\rightarrow \mathbb{R}^p$ ...
0
votes
1answer
23 views

Preimage of principal bundle under equivariant map

Let $M$ be a manifold, $G,H$ be some Lie groups, $\sigma:G\to H$ be a Lie group homomorphism, $K\subset H$ a maximal compact subgroup of $H$ and $\tilde K:=\sigma^{-1}(K)\subset G$ . Let further $\...
2
votes
1answer
25 views

Divergent Curves and Complete Manifolds

I'm working on a problem in do Carmo's Riemannian geometry book (chapter 7, problem 5). He states that a divergent curve on a noncompact Riemannian manifold $M$ is a curve $\alpha: [0, \infty) \to M$ ...
2
votes
2answers
36 views

When are embeddings into Euclidean space unique up to ambient isometry?

Suppose I have a Riemannian smooth manifold $M$ and a smooth isometric embedding $M \hookrightarrow \mathbb{R}^n$. Is this embedding necessarily unique up to some isometry of $\mathbb{R}^n$? If not, ...
3
votes
3answers
85 views

Differential Geometry for General Relativity

I'm going to start self-studying General Relativity from Sean Caroll's Spacetime and Geometry: An Introduction to General Relativity. I'd like to have a textbook on Differential Geometry/Calculus on ...
2
votes
0answers
44 views

nef Line bundles over Kähler manifolds

I am trying to understand a particular property of the first Chern class of a nef line bundle over a Kähler manifold. We know in general, let $X$ be a complete complex projective variety, and $L$ a ...
4
votes
1answer
57 views

Abstract algebraic definition of dual tangent spaces

I know that if $(M,\mathcal{A})$ is a smooth manifold, the dual tangent space at $p\in M$ can be defined as $$ T^*_pM=I_p/I_p^2, $$ where $I_p$ is the ideal of the ring $C^\infty(M)$ consisting of ...
1
vote
1answer
50 views

Topology, atlas, smooth manifold

Let $X$ be a set, $n\in\mathbb{N}$ and $((U_i,\phi_i))_i$ a family of subsets $U_i\subseteq X$ with injective functions $\phi_i: U_i\to\mathbb{R}^n$, which hold the following conditions: $\...
0
votes
1answer
26 views

Particle motion and frenet frame

I am given that $\hat{t}=\dfrac{\hat{x}+y'\hat{y}}{\sqrt{1+y'^2}}$ and $\hat{n}=\dfrac{y'\hat{x}-\hat{y}}{\sqrt{1+y'^2}}$ are the tangential and normal vector in frenet frame. We are considering only ...