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)

0
votes
0answers
8 views

What 's conditions on open set related to connected neighborhood of boundary

I have a question: Suppose $D$ is an open set in $\mathbb{R^n}$ and topological boundary $bD$ is an embedded submanifold of $\mathbb{R^n}$. For each $p\in bD$, we want to have an open neighborhood ...
0
votes
0answers
19 views

Invariant in geodesic

What in general is invariant in geodesic in terms of parameters $u$ and $v$ ( or functions on which they depend) and their derivatives in integrated form? For a surface of revolution, Clairaut's ...
0
votes
0answers
13 views

Definition of a lipschitz 1-form on a manifold

What is the definition of a Lipschitz-regular 1-form on a riemannian manifold?
3
votes
0answers
25 views

How “far” a differential form is from an exterior product

Consider two differential manifolds $X$ and $Y$. Consider now a differential form (of any order) $\omega$ on $X\times Y$. The easiest example is taking $\omega=\xi\wedge\eta$, where $\xi$ is a ...
1
vote
2answers
31 views

The differential $\text d F_p$ is injective iff the pullback $F_p^*$ is surjective.

I'm trying to prove the following claim: Let $F\colon M \to N$ be a differentiable application beetween $C^\infty$ manifolds. Then the differential $\text dF_p\colon T_p M \to T_{F(p)}N$ is ...
2
votes
1answer
28 views

differential form identity and permutations

If $t^1,...,t^k$ are the coordinates of a k-cube. Then apparently $$dt^{\sigma(1)} \wedge \ldots \wedge dt^{\sigma(k)}= (\operatorname{sgn} (\sigma)) dt^1 \wedge dt^k $$ I cannot see how this ...
1
vote
1answer
14 views

contraction identity on $k$-forms

$i_\mathbb{X} \omega $ is the contraction of $\omega$ with respect to $\mathbb{X}$. In my notes it is stated that $i_\hat{\mathbb{X}} dx = dx(\hat{\mathbb{X_t}})$. I cannot see how this fits the ...
0
votes
1answer
24 views

Start of the proof of the Poincaré Lemma

Let $\hat{\Phi}:U \rightarrow U$ be a one-parameter family of diffeomorphisms defined for $\ 0 < t \leq 1$. Let $\beta \in \Omega^k(U) $ be a closed k-form. Suppose that $\hat{\Phi}^{*}_1 \beta = ...
0
votes
0answers
22 views

General expression of smooth sections of tensor bundles.

On page 317 of John Lee's Smooth Manifolds it's said that if $(x^i)$ are local coordinates on a smooth manifold $M$, then sections of the tensor bundle $T^kT^*M=\bigsqcup_{p\in M}T^k(T^*_pM)$ over a ...
1
vote
3answers
53 views

Showing that a Unit Speed Curve is a Circle.

In my recent differential geometry tutorial, we were given the question: Given the unit speed curve, $$\boldsymbol{r}(s)=\left(\frac{4}{5}\cos(s),1-\sin(s),-\frac{3}{5}\cos(s)\right)$$ show that ...
2
votes
1answer
41 views

Book on advanced Hodge Theory

I'm looking for a book on advanced real Hodge Theory. I finished working through Frank Warner's Foundations of Differentiable Manifolds and Lie Groups, which ends with the Hodge Decomposition,the ...
1
vote
0answers
22 views

Is Hodge star operation can be understood as contraction after tensor product of a $p$-form with the volume element?

By defintion, the Hodge star of a $p$-form $\omega_{a_1\cdots a_p}$ on a $n$-dimensional manifold is given by $*\omega_{b_1\cdots b_{n-p}}=\frac{1}{p!}\omega^{a_1\cdots a_p}\epsilon_{a_1\cdots ...
-2
votes
0answers
10 views

Intersection between hyperplane with in a simplex [on hold]

If we contract a (n-1)d hyperplane with a n-simplex, then what is maximum number of intersection points with the egdes of the simplex and the hyperplane ? For, if we draw a line within a 2-simplex ...
1
vote
0answers
38 views

geometrical consequences of nonpositive or negative Ricci curvature.

well,that's pretty much the question. I'd like to know if somebody could point me out if there's any geometrical implications following an upper bound on the Ricci curvature on a riemannian manifold. ...
2
votes
0answers
14 views

The invariance of the Ricci tensor under diffeomorphisms and its non-ellipticity.

Consider $(M,g)$ a compact Riemannian manifold. When viewed as a second order (non-linear) differential operator $$ \text{Ric} : C^{\infty}(\text{Sym}^2_+T^*M) \to C^{\infty}(\text{Sym}^2T^*M), $$ the ...
2
votes
0answers
41 views

Ovals of constant $ k_g$ on constant $K$ surfaces

Prove that: Constant geodesic curvature lines on constant Gauss curvature surfaces are closed Ovals/Loops. Find perimeter/length of this Oval/Loop in terms of $ k_g$ and $K$ I believe the proof ...
0
votes
0answers
23 views

What is the notation for pull-back and push-forward of an exponential map?

So there is a nice notation for a one-parameter group of transformations $\Phi_t$ corresponding to its infinitesimal generator $\boldsymbol X$: $$\Phi_t = \exp \left(t \boldsymbol X \right)$$ But ...
1
vote
0answers
42 views

Calabi-Yau manifolds and immersion in real space [on hold]

I'm reading some papers how to test extra dimensions in LHC experiments and they suggests CY manifolds as starting point. Is it possible that accelerator itself is made in higher-dimensional geometry ...
0
votes
2answers
38 views

Differential identity and wedge products

Apparently $dx^{i_1} \wedge ... \wedge dx^{i_k}=d(x^{i_1}dx^{i_2}\wedge ... \wedge dx^{i_k})$ which I cannot see proved anywhere in my notes. It just stated as if it is obvious which I don't believe ...
2
votes
2answers
80 views

Geometric Product

I have a problem with the geometric product: In my book the unit trivector is defined like this: $(e_{1}e_{2})e_{3}=e_{1}e_{2}e_{3}$ But that would mean $(e_{1}e_{2})e_{3}= (e_{1} \wedge e_{2})\cdot ...
2
votes
2answers
63 views

Part of proof that $d^2\omega=0$

The following comes from the proof in differentiable manifolds that $d^2\omega=0$. Let $f$ belong to the set of $0$-forms. From definition I have that $\displaystyle df = \frac{\partial f}{\partial ...
0
votes
1answer
31 views

Define a parametrized curve $\beta:(a,b)\rightarrow\mathbb R^3$ by $\beta(t)=\frac{d\gamma(t)}{dt}$

Let $\gamma:(a,b)\rightarrow\mathbb R^3$ be a unit speed space curve with non-vanishing curvature $\kappa(t)\neq0$. Define another parametrized curve $\beta:(a,b)\rightarrow\mathbb R^3$ by ...
2
votes
0answers
38 views

tangent spaces of manifols

I have two manifolds $E_{n}=\{([0:x_{1}:x_{2}],[y_{1}: y_{2}])\in \mathbb{C}\mathbb{P}^{2}\times \mathbb{C}\mathbb{P}^{1}\}$ e $V_{n}=\{([x_{0}:0:x_{2}],[y_{1}: y_{2}])\in ...
0
votes
1answer
33 views

Non-punctual Boundary

In the book of Bill Thurston, Three dimensional geometry and topology, there is an exercise to show torus can be partitioned into 7 countries, each on one piece and has common (non-punctual) ...
2
votes
1answer
18 views

Number of intersections of two closed loops on a genus zero surface

I have stumbled onto the following fact and I am quite helpless in seeing why this is true (although I can agree intuitively). Let $M$ be a surface of genus zero (open or closed, with or without ...
0
votes
1answer
39 views

Tangent bundle of a tubular neighborhood

Let $N \to X$ be normal bundle of a submanifold $X$ of $Y$. How can I prove that $TN|_{TX}$ is isomorphic to the normal bundle of the inclusion $TX\to TY$? And why this vector bundle is isomorphic to ...
0
votes
0answers
42 views

Arc length for a function $f:\mathbb{R}^2 \to \mathbb{R}^2.$

Assume $f:\mathbb{R}^2 \to \mathbb{R}^2$ is $C^1.$ Is there a formula for the length of the subset of $\mathbb{R}^2$ given by $$ \{f(x,y) \in \mathbb{R}^2:a_1\leq x\leq b_1, a_2\leq y \leq b_2\} ? $$ ...
8
votes
0answers
126 views
+200

Symmetric and wedge product in algebra and differential geometry

I have been struggling with this issue for a while (and asked a similar question here), but still not found a satisfying answer. The question boils down to: which is the correct identity? $dx \, dy ...
0
votes
0answers
33 views

Topology of biological compartments

In the field of cell biology, there is a general sub-field concerned with the topology of organelle membranes, and a key focus remains on how these dynamic membranes deform and interact with cellular ...
1
vote
1answer
43 views

Twisting with a degree negative line bundle

Let $X$ be a Riemann surface. Let $M_1$ and $M_2$ be two holomorphic bundles on $X$. Does injectivity of $h^0(M_1)\to h^0(M_2)$ imply $h^0(M_1\otimes L)\to h^0(M_2\otimes L)$ is injective? Where $L$ ...
3
votes
1answer
62 views

Can we measure how close a vector bundle is to being trivial?

For a vector bundle $E$, I will denote the maximum number of linearly independent global sections of $E$ by $\eta(E)$. We have $\eta(E) \in \{0, 1, \dots, \operatorname{rank}(E)\}$ and $\eta(E) = ...
1
vote
0answers
41 views

Manifold characteristics in terms of Riemannian metric

I wonder what characteristics of Riemannian manifold can be expressed in terms of metric? Are there any results similar to Gauss–Bonnet theorem? Does the Riemannian metric give any information about ...
0
votes
1answer
29 views

Tangent bundle of the 2-sphere

I'm reading through Tu's Introduction to manifolds and today I learned about tangent bundles and vector bundles. I was surprised to learn that $TS^2$, tangent bundle of the 2-sphere, isn't trivial ...
0
votes
0answers
26 views

Parallel Transport and Christoffel Symbol

For two near by points in General Theory of Relativity. The change in the vector components when parallel transported is given by Now, since the parallel transport change must depend on the path ...
4
votes
2answers
91 views

How to prove that all smooth vector bundles on a given vector bundle are the pull back of a vector bundle on the base

Recently, during a conversation, I heard about the result (previously mentioned also here on MO), whose statement is reported below. Not having the specific background necessary to reconstruct a proof ...
1
vote
2answers
31 views

Definition of a coordinate vector bundle

Consider the following definition of a coordinate vector bundle. Let $M$ be a smooth manifold of dimension $m$, and $\{(f, U_f)\}$ an atlas of compatible charts for $M$. A smooth coordinate ...
0
votes
0answers
18 views

Laplacian in arbitrary spherical coordinates?

assume that $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is a function that just depends on the radius vector $r$, so no angular dependence(!). Can we say what $\Delta f$ is in arbitrary coordinates $r ...
1
vote
0answers
25 views

Can the closure of a geodesic chart of a manifold without boundary (and with bounded geometry) be defined as a compact manifold with boundary?

Let $(M,g)$ be a $d$-dimensional smooth, riemannian manifold without boundary, which is geodesically complete, connected, has positive injectivity radius and a bounded geometry (i.e. all derivatives ...
0
votes
0answers
39 views

Pushforward of a volume form

Let $X$ be a complex projective manifold with semi-ample line bundle $ K_X$ . Assume that $f: X\to X_{can}\subset \mathbb CP^N$ , and $f^{-1}(s)$ is nonsingular fibre, then I am looking for a proof ...
1
vote
1answer
32 views

Variable Pitch Helices

Is it necessary for a helix to have constant pitch? If it is not so, what would be equation of a variable pitch helix?
0
votes
1answer
29 views

Definite Integral theorem validity :- $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds$?

Can we write $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds\tag 1$ ? In other words, is this result valid? If so, could you help me to get the proof it NB :: ...
1
vote
0answers
21 views

Invariant characterization of vector bundles associated to a principal bundle?

I have two related questions. Suppose I have a principal $G$-bundle $P\xrightarrow{\pi} M$. The usual construction of an associated vector bundle goes as follows. Fix some representation $\rho : ...
2
votes
2answers
43 views

Exercise on left invariant 2-forms

I'm trying to solve a problem about the Lie group of transformation over $\mathbb{R}$: \begin{equation} x\mapsto ax+b, \end{equation} where $a,b\in\mathbb{R}, a\neq 0.$ I'm asked to find the space of ...
0
votes
0answers
34 views

Pseudo-scalar product on Manifold

I'm trying to study the Semi-Riemannian Manifold and the relativity (I use the book Semi-Riemannian Manifold- O'Neill). But I don't understand the following thing: In a Semi-Riemannnian Manifold, I ...
3
votes
1answer
39 views

Do Anosov flows exist on two dimensional compact manifolds?

Question: Let $M$ denote a two-dimensional, smooth, compact Riemannian manifold (without boundary). If $\phi:\mathbb{R}\times M \rightarrow M$ is a smooth flow, then prove that $\phi$ is not an ...
1
vote
0answers
12 views

Ribbon Surfaces and Legendrian Graphs on Contact 3-manifolds.

Let $M=(M, \xi)$ be a contact 3-manifold. I am trying to show that every Legendrian graph L (i.e., a graph embedded in $M$ so that it is everywhere-tangent to the contact planes) admits a ribbon ...
-2
votes
0answers
43 views

Kahler condition [closed]

Let $(M,\omega) $be a Kahler manifold. Why is the Kahler condition $$d \omega = 0$$equivalent to: $$\partial_kg_{i\bar j} = \partial_ ig_{k\bar j}$$ for all $i; j; k$?
0
votes
1answer
33 views

Sign of first chern class with some conditions

Let $X$ be a compact Kahler algebraic variety which $K_M$ is big and nef, and $Kod(X)=dimX$ then why the first chern class $c_1(M)$ is negative or zero . I don't undrestand kawamata's theorem in this ...
3
votes
1answer
41 views

Differential of the inversion of Lie group

Let $G$ be a Lie group and $\iota \colon G\to G$ denote the inversion. If $e$ is the identity of $G$, prove that: $$\text d \iota _e = -\text {id} _{T_e G}.$$ I understand that the differential at $e$ ...
1
vote
1answer
28 views

2-forms on $S^2$

I've read that the group $H^2_{dR}(S^2)=\mathbb{R}$. If I'm not wrong, this implies that one can build closed 2-forms that are not exact. Can somebody show me an example, please? Thanks!