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)

37
votes
0answers
926 views

Differential forms on fuzzy manifolds

This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me. Question In the usual setting of open subsets of ...
26
votes
0answers
782 views

Computing the Chern-Simons invariant of SO(3)

I am an undergraduate learning about gauge theory and I have been tasked with working through the two examples given on pages 65 and 66 of "Characteristic forms and geometric invariants" by Chern and ...
14
votes
0answers
922 views

What is magical about Cartan's magic formula?

Why is Cartan's magic formula $$\mathscr{L}_X\omega = i_Xd\omega + d(i_X\omega)$$ called "magic"? Should it be considered a highly surprising result? Does it "magically" prove several other ...
11
votes
0answers
217 views

Kähler Geodesics

Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric $$\begin{pmatrix} ...
10
votes
0answers
168 views

A short question on shriek maps

This should be easy but I don't quite see it. Let $M^m, N^n, X^d$ be compact, connected and oriented smooth manifolds. Let also $f:M\rightarrow X$ and $g:N\rightarrow X$ be transverse smooth maps. ...
10
votes
0answers
415 views

A Way to make the following “proof” of the Hairy Ball Theorem rigorous?

I plan on giving a talk soon to undergraduates and I'd like to talk about the hairy ball theorem during the talk. I was trying to think of some sort of visually intuitive proof of this fact. (I ...
8
votes
0answers
122 views

Does Stokes' Theorem hold on spaces with singular points?

I have come across the question whether Stokes' theorem holds also on orbifolds. Let us take the simple case of $T^2/Z_2$ with a one-form $A$, then the question becomes: For a region $\Gamma$ with ...
8
votes
0answers
157 views

algebraic $1$-forms vs analytic $1$-forms

First let's fix some definitions: Definitions: Complex manifold (of dimension n): Is a locally ringed space $(X,\mathscr F)$, where there is an open cover $\bigcup_{i\in I} U_i=X$ such that ...
8
votes
0answers
219 views

De Rham Cohomology of $M \times \mathbb{S}^1$

Let $M$ be a closed (compact, without boundary) $m$-dimensional manifold. I want to prove that $H^{k+1}(M \times \mathbb{S}^1) = H^k(M) \oplus H^{k+1}(M)$. ($H^k$ is the $k$-th De Rham cohomology ...
8
votes
0answers
134 views

Different notions of isometry for Riemannian 2-manifolds

There are two notions of isometry between Riemannian 2-manifolds: a distance-preserving map $f$ with $d(x,y) = d(f(x),f(y))$ and a "metric-preserving" map $f$ with $I(x) = I(f(x))$ ($I(x)$ being ...
8
votes
0answers
290 views

Stokes theorem on Lipschitz-manifolds?

I was wondering if Stokes' theorem could be formulated in a setting which could be easily applied in situations where the traditional form cannot, such as on manifolds with corners like a rectangle or ...
8
votes
0answers
157 views

Curvature 0 and involutive horizontal distributions

I am trying to check why curvature 0 implies that the horizontal distribution is involutive. Let $\pi:P\to U$ be a principal $G:=GL_n$ bundle. Assume that $P$ is trivial and $\pi$ admits a section. ...
7
votes
0answers
87 views

Characterization of the exterior derivative $d$

In the paper Natural Operations on Differential Forms, the author R. Palais shows that the exterior derivative $d$ is characterized as the unique "natural" linear map from $\Phi^p$ to $\Phi^{p+1}$ ...
7
votes
0answers
136 views

If a thread is pulled out of a floating blob of water, must the thread be tangent to the surface of the blob at some point?

My motivation is the recent question I just answered, and my answer use too many hypothesis that I considered superfluous: Always "one double root" between "no root" and "at ...
7
votes
0answers
60 views

Connection in fibre bundle from discontinuous group action

I am trying to understand connections in fibre bundles. I thought of the following problem: Let $\Gamma$ be the discrete group generated by \begin{pmatrix} 1 & 3 & 0 \\ 0 & 1 & 0 \\ ...
7
votes
0answers
108 views

Two definitions of $H^1(\partial\Omega)$, one using charts and one use tangential gradients

Let $\Omega$ be a bounded Lipschitz domain with boundary $\partial\Omega$. There are two ways to define a space $H^1(\partial\Omega)$: By using charts, we can define $H^1(\partial\Omega)$ to ...
7
votes
0answers
323 views

Trying to Understand Lefschetz Pencils

I'm reading on Lefschetz pencils, and I'm trying to understand condition ii) better, tho I would appreciate insights on condition i), and in general. A Lefschetz pencil on a $4$-manifold $X$ is a ...
7
votes
0answers
280 views

Hodge-Star-Operator on arbitrary oriented basis

Assume that $V$ is oriented finite dimensional vectorspace with dimension $n$, $g \in T^0_2(V)$ a given symmetric and nondegenerate tensor. Let $\mu$ be the corresponding volume element of $V$. ...
7
votes
0answers
124 views

Translating a passage of a paper by L. Bérard Bergery

I am currently studying the following paper on Einstein manifolds: L. Bérard Bergery, Sur de nouvelles variétés riemanniennes d'Einstein, Inst. Elie Cartan, Univ. Nancy №6, 1-60 (1983). I have ...
7
votes
0answers
129 views

The rigorization of naive geometry angles and length

There are a number of claims from elementary school that I just remembered I don't actually mathematically know. Let's start with some specific examples and perhaps the rigorization will inspire me ...
7
votes
0answers
354 views

Torsion in two dimensions?

This question is about the notion of a connection with torsion in differential geometry, i.e., a connection that is not Levi-Civita. (It's not about the torsion of a curve in three dimensions.) ...
7
votes
0answers
109 views

Dimension of diffeomorphism groups preserving some $2$-tensor.

For a finite-dimensional smooth manifold $M$, let $\mathrm{Diff}(M)$ be its diffeomorphism group. Suppose we are given a $2$-tensor $\mathcal{K}$ on $M$, and let $$\mathrm{Diff}_{~\mathcal{K}}(M) = ...
7
votes
0answers
285 views

Laplace-Beltrami Operator for Euclidean Space

Consider the space $\mathbb{R}^n$ and let $x_1,\ldots, x_n$ be the coordinates. Fix the orientation $dx_1\wedge dx_2\ldots\wedge dx_n$. Let $E^p$ denote the space of smooth $p$ forms and let $d$ ...
7
votes
0answers
321 views

Curvature and connections in principal G-bundles

Let $E \rightarrow F$ be a principal $G$-bundle. Let $\alpha$ be a connection 1-form with values in the Lie algebra of $G$. Let $\omega$ denote the curvature 2-form of connection $\alpha$. We know ...
7
votes
0answers
278 views

Riemannian Immersions into Euclidean Space?

The Whitney embedding theorem states that any smooth manifold can be embedded in Euclidean space. In the Riemannian setting this naturally leads to the question whether this can be done in such a way ...
6
votes
0answers
51 views

Differentiation on Manifolds Basics

I'm having some real trouble comprehending integral curves and Lie derivatives on a Manifold. I will write out my understanding and ask the questions below. For a vector field $X$ on smooth manifold ...
6
votes
0answers
65 views

Computing Hodge numbers of a complete intersection

The situation is this: I have a 5-dimensional irreducible projective variety $Y$ embedded in $\mathbb P^{13}$. This variety is singular, the singularities being a disjoint union of two curves. I have ...
6
votes
0answers
94 views

Sobolev space on $M \times [0,\infty)$, $M$ compact closed manifold

Consider a manifold of the form $M \times [0,\infty)$, where $M$ is a closed compact manifold without boundary. So $M \times [0,\infty)$ is a semi-infinite cylinder. I want to know information about ...
6
votes
0answers
64 views

A property processed by a special vector field.

Let $Y$ be a vector field on $\mathbb R^n$ (or any Riemannian manifold $(M, g)$). When will we know that $$Y =\nabla_X X$$ for some other vector fields $X$? Or more precisely, if $Y$ does satisfy ...
6
votes
0answers
85 views

What is a moduli space for a differential geometer?

A moduli space is a set that parametrizes objects with a fixed property and that is endowed with a particular structure. This should be an intuitive and general definition of what a moduli space is. ...
6
votes
0answers
124 views

Helices in Lorentz-Minkowski space $\Bbb L^3$.

Context: Consider the Lorentz-Minkowski space $\Bbb L^3$, with the metric: $$ds^2 = dx^2 + dy^2 - dz^2$$ (which I'll denote just by $\langle \cdot, \cdot \rangle$) If $\alpha$ is spacelike and ${\bf ...
6
votes
0answers
151 views

Which manifolds are zero sets of $\mathbb R^n$ valued maps

If $M$ is a smooth manifold, then any framed submanifold $N$ is the preimage $f^{-1}(y) $ for a smooth sphere-valued map $f$ transversal to $y$, with the framing of the normal bundle induced by $f$. ...
6
votes
0answers
87 views

What is the generalization of Gauss's Theorem to a manifold?

In a (pseudo-)Riemannian manifold with constant basis vectors, one certainly has that the integral of the divergence of a tensor field $T$ over a submanifold $\Omega$ is equal to the integral over the ...
6
votes
0answers
47 views

Two definitions of real flag manifolds: do they coincide?

Let $G$ be a real semisimple Lie group with finite center. Definition 1 A real flag manifold is a homogeneous space $G/Q$ where $Q$ is a parabolic subgroup of $G$. Definition 2 Let $K$ be a ...
6
votes
0answers
145 views

determinant of the Fubini-Study metric

Is there any easy way to compute the determinant of the Fubini-Study metric, given by: $g_{\alpha\bar{\beta}}=\frac{1}{1+\bar{z}z}\left(\delta_{\alpha\bar{\beta}}-\frac{\bar{z}_\alpha ...
6
votes
0answers
316 views

Surface integral for a scalar function defined on a discrete surface

Imagine a polyhedral, discrete surface embedded in $\mathbb{R}^3$. Its faces are all triangles. For each vertex, one can compute the discrete mean and Gaussian curvatures and evaluate the sum of ...
6
votes
0answers
77 views

How are boundary conditions formally captured by the jet bundle approach to differential equations?

In the jet bundle approach to differential equations https://en.wikipedia.org/wiki/Jet_bundle#Partial_differential_equations one identifies the equation with the set of a solution of the ...
6
votes
0answers
140 views

1-form with positive integral over a path

Given any smooth path $\gamma$ on a manifold $M$, when can we construct a 1-form $\omega \in \Omega^1(M)$ so that $\int_{\gamma} \omega > 0$? It is easy to see that such $\omega$ exists if the path ...
6
votes
0answers
111 views

When is There a Solution to “Pullback Equation” of Differential Forms

All: Let $f: M \to N$ be a smooth map between manifolds, and let $w$ be a $1$-form on $M$. Under what conditions is there a $1$-form $z$ defined on $N$ so that $w=f^*z$, i.e., so that $w$ is the ...
6
votes
0answers
506 views

Poincaré-Hopf theorem and its applications

I'm reading the basics of differential topology to try to understand the Poincaré-Hopf theorem, its proof and its applications. My plan is as follows: 1) Study transversality: its homotopy stability ...
6
votes
0answers
154 views

“Natural” constructions of tensor fields from tensor fields on a manifold

This question begins is related to this question on physics.SE Uniqueness of Riemann Curvature Tensor, which asks roughly "what tensors can we make locally out of just the metric tensor? We can ...
6
votes
0answers
279 views

sheaf of differential forms - tangent sheaf [Hartshorne]

I'm reading section 8 Differentials of chapter 2 in Hartshorne. It's is extremely hard to me to understand the nature of the definitions: module of relative differential forms - sheaf of relative ...
6
votes
0answers
615 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 ...
6
votes
0answers
86 views

Do balls optimize the boundary area for a fixed volume?

I recently saw this question, asking if the circle is the planar figure which has the least perimeter given the area. As far as I know, this is a classical problem, and the answer is affirmative. I am ...
6
votes
0answers
147 views

Isomorphism between spaces of sections.

Let $M$ be a compact manifold and let $E_i \xrightarrow{\Large \pi_i} M$, $i = 1, 2$, be two (real or complex) vector bundles of the same rank $k$ over $M$. Assume we have metrics $g_1, g_2$ on $E_1$, ...
6
votes
0answers
104 views

Identification of integration on smooth chains with ordinary integration

Let $M$ be a smooth oriented $n$-dimensional manifold and denote by $A \in H_n(M;\mathbb{Z})$ the fundamental class of $M$ (a generator of singular homology consistent with the orientation of $M$). ...
6
votes
0answers
359 views

Characterization of gradient vector fields

Let $V$ be a vector field on a smooth manifold $M$. Are there nice conditions under which there exists a (Riemannian) metric on $M$ such that $V$ is the gradient of some smooth function on $M$? One ...
6
votes
0answers
314 views

Orientability of the total space of a vector bundle over an oriented manifold

Let $M$ be a (smooth) manifold of dimension $n$, and let $\pi : E \to M$ be a (smooth) vector bundle of rank $r$. If I choose a connection on $E$, then I obtain a decomposition of $T E$ as $V E ...
6
votes
0answers
433 views

Higher-order derivatives in manifolds

If $E, F$ are real finite dimensional vector spaces and $\mu\colon E \to F$, we can speak of a (total) derivative of $\mu$ in Fréchet sense: $D\mu$, if it exists, is the unique mapping from $E$ to ...
6
votes
0answers
371 views

Lie group action from infinitesimal action

I would like to ask, how to deduce a Lie group action, from infinitesimal action of its Lie algebra (the so called Lie-Palais theorem). More precisely, given a differential manifold $M$ and a Lie ...