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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9
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237 views

checking if a 2-form is exact

Consider the 2-form $$\sigma=\frac{x_1 dx_2 \wedge dx_3 + x_2dx_3\wedge dx_1+ x_3 dx_1 \wedge dx_2}{(x_1^2+x_2^2+x_3^2)^{3/2}}.$$ I need to show if it is exact or not. Suppose it is exact, then there ...
8
votes
1answer
635 views

Showing a subset of the torus is dense

Let $T^2\subset\mathbb{C}^2$ denote the (usual) torus. Let $a\in\mathbb{R}$ be an irrational number and define a map $f(t)=(e^{2\pi it}, e^{2\pi i at})$. Prove that: (a) $f$ is injective (b) $f$ is ...
7
votes
3answers
445 views

Are all manifolds in the usual sense also “vector manifolds”?

In geometric calculus, there is a concept of a vector manifold where the points are considered vectors in a general geometric algebra (a vector space with vector multiplication) which can then be ...
7
votes
2answers
458 views

Trace of a bilinear form?

I'm just a beginner of differential geometry, so please forgive me if this is nothing but a silly question or I'm making a critical conceptual mistake. Let $\mathrm{I\!I}(X, Y)$ be the second ...
6
votes
1answer
163 views

The continuity of multivariable function

$F$ is a function on $\mathbb R^n$ such that for every smooth curve $\gamma:[0,1] \rightarrow \mathbb R^n, \gamma(0)=0 $, we have $\mathop {\lim }\limits_{t \to 0} F(\gamma (t)) = 0$, is it necessary ...
6
votes
4answers
419 views

Reference request: Vector bundles and line bundles etc.

I am interested in learning algebraic geometry and I talked to one of my professors today (who is, in part, an algebraic geometer) and he recommended I understand the analytic analogue of the ideas in ...
5
votes
1answer
100 views

Submanifold given by an open immersion

I was wondering if the following is true: Let $M,N$ be two manifolds such that $\dim M\leq \dim N$ and $f:M\rightarrow N$ an smooth immersion. Assume that for any open set $U\subset M$, ...
5
votes
2answers
160 views

Compatibility of a connection and metric

Every Riemannian manifold admits a metric connection. Suppose $M $ is a manifold and $\nabla $ is an arbitrary connection on the tangent bundle. Does $M$ necessarily admit a metric such that $\nabla$ ...
5
votes
0answers
230 views

Tangent bundle of a quotient by a proper action

Given a compact group $G$ acting freely on a manifold $X$, is there a "nice" way to describe the tangent bundle of the quotient $X/G$ (when it is a manifold)? In the case the group $G$ is finite, or ...
5
votes
1answer
371 views

What are some isometries of $S^2$ without fixed points?

This spherical geometry question involves isometries. I am particularly looking for isometries with no fixed points.
5
votes
1answer
203 views

A tensor calculus problem

If the relation $a_{ij}u^iu^j=0$ holds for all vectors $u^i$ such that $u^i\lambda_i=0$ where $\lambda_i$ is a given covariant vector, show that $$a_{ij}+a_{ji}=\lambda_iv_j+\lambda_j v_i$$ where ...
5
votes
3answers
338 views

Geodesics of a “diagonal” metric

Are there any relations that exist to simplify Christoffel symbols/connection coefficients for a diagonal metric which has the same function of the coordinates at each entry? In other words, I have a ...
5
votes
1answer
604 views

The winding number and index of curve

If $\gamma $ is a smooth closed curve in $\mathbb R^2-\{0\}$, I want to know whether the winding number of $\gamma $ about $0$, i.e., $\frac{1}{{2\pi i}}\int\limits_\gamma {\frac{1}{z}} dz$ is equal ...
5
votes
4answers
521 views

What does the symbol $\operatorname{Tr}$ in the Yang-Mills action mean?

I find that many authors write the Yang-Mills action as follows: $$\mathcal{J}= \int \operatorname{Tr}(F \wedge \star F).$$ I have yet to find a formal description of the symbol $\operatorname{Tr}$ ...
5
votes
1answer
350 views

Tautological vector bundle over $G_1(\mathbb{R^2})$ isomorphic to the Möbius bundle

Let $V$ be a finite dimensional vector space, and let $G_k(V)$ be the Grassmannian of $k$-dimensional subspaces of $V$. Let $T$ be the disjoint union of all these $k$-dimensional subspaces and ...
5
votes
2answers
518 views

Gradient in differential geometry

I am a graduate student in physics trying to learn differential geometry on my own, out of a book written by Fecko. He defines the gradient of a function as: $ \nabla f = \sharp_g df = g^{-1}(df, ...
5
votes
1answer
460 views

Degree of Gauss map equal to half the Euler characteristic and Poincaré-Hopf

The Poincaré-Hopf theorem states that for a smooth compact $m$-manifold $M$ without boundary and a vector field $X\in\operatorname{Vect}(M)$ of $M$ with only isolated zeroes we have the equality ...
4
votes
2answers
167 views

Complex and Kähler-manifolds

I was woundering if anyone knows any good references about Kähler and complex manifolds? I'm studying supergravity theories and for the simpelest N=1 supergravity we'll get these. Now in the ...
4
votes
1answer
88 views

Diffeomorphisms preserving harmonic functions

I'm looking for smooth maps $ \Bbb R^n \to \Bbb R^n $ with the property that, whenever $ h $ is a harmonic function ($\Delta h=0$), $ h\circ f $ is also harmonic. Is there a nice characterization of ...
4
votes
3answers
159 views

The interior product and the isomorphism $\bigwedge^k(V^*)\otimes\bigwedge^n(V)\cong\bigwedge^{n-k}(V)$

Let $V$ be an $n$-dimensional vector space. According to Wikipedia, there is an isomorphism $\bigwedge^k(V^*)\otimes\bigwedge^n(V)\cong\bigwedge^{n-k}(V)$. The explanation is that for $\alpha \in ...
4
votes
0answers
163 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 ...
4
votes
2answers
153 views

Orientations on Manifold

This is a very basic definition of orientable and very basic example 20.5 however ı could not understand definition in an good way so ı want you to explain my green writing please :) and my example ...
4
votes
2answers
251 views

Does the curvature determine the metric?

Here I asked the question whether the curvature deterined the metric. Since I am unfortunately completely new to Riemannian geometry, I wanted to ask, if somebody could give and explain a concrete ...
4
votes
2answers
263 views

The cone is not immersed in $\mathbb{R}^3$

The problem is to show that the cone $ z^2= x^2+y^2$ is not an immersed smooth manifold in $\mathbb{R}^3$.
3
votes
3answers
81 views

Commutator of Vector Fields

Q: Given the vector fields $A=x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}$, $B=x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}$ Calculate the commutator $\left[A,B\right]$. ...
3
votes
2answers
97 views

Line bundle over $S^2$

I'm trying to study line bundle over $S^2$. In this post was outlined the method based on clutching functions. But now I'm interesting in another approach. For the sphere there is two maps : upper ...
3
votes
1answer
282 views

Calculate the curvature of a parametrized curve

I have started to study differential geometry and have some questions about an exercise which is probably not very difficult. Exercise: Let $\gamma: I \rightarrow\mathbb{R}^{2}$ be a regular curve, ...
3
votes
2answers
175 views

Curvature of a regular parametrization

Prove that if $\mu: [a,b] \to \mathbb{R}^n$ is a regular parametrization of a curve then the curvature at $\mu(t)$ is given by: $$\kappa(t) = ...
3
votes
2answers
546 views

Real line bundle smoothly isomorphic to Möbius bundle

I am reading Lee's Introduction to Smooth Manifolds and got stuck on the problem 5.6. The question is written here, question 1. (There is a typo in the question. The last sentence should be "Show that ...
3
votes
4answers
605 views

Tangent Bundle on S^3

how to show T(S^3) isomorphic to S^3 cross R^3? so can I say it for every odd dimension?I have shown it for n=1
2
votes
2answers
138 views

What does $dx$ mean in differential form?

This question relates to this post. From what I know in calculus and standard analysis, strictly speaking, there is no meaning of $dx$. It only makes sense when combining with another $d$, e.g. ...
2
votes
1answer
41 views

Parallel Curve of Regular Plane Curves

Let $\gamma$ be a regular plane curve and let $\lambda$ be a constant. The parallel curve $\gamma^\lambda$ of $\gamma$ is defined as $$\gamma^\lambda(t)=\gamma(t)+(\lambda)n_s(t),$$ where $n_s$ is the ...
2
votes
2answers
399 views

How to measure the distance between two cities in the map by knowing latitude point and longitude point of them?

I want to measure the distance between two points in a map. For example between London and Moscow by knowing that the latitude point and longitude point of them. ...
2
votes
1answer
121 views

Gradient of a functional

Given a compact manifold with a Riemannian metric $g$, we define the total scalar curvature by $$E(g)=\int_M RdV$$ Let us consider the first variation of $E$ under an arbitrary change of metric. We ...
2
votes
1answer
303 views

Quotient theorem for tensors

Can somebody please explain to me how the following statement is true? The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
2
votes
1answer
1k views

Calculating mean and Gaussian curvature

I am stuck on this question from a tutorial sheet I am going through. Compute the mean and Gaussian curvature of a surface in $\mathbb{R}^3$ that is given by $z=f(x)+g(y)$ for some good functions ...
2
votes
4answers
1k views

Problems that differential geometry solves

Recently, I've been studying a course in differential geometry. Some keywords include (differentiable) manifold, atlas, (co)tangent space, vector field, integral curve, lie derivative, lie bracket, ...
2
votes
2answers
762 views

Taking trace of vector valued differential forms

Can anyone kindly give some reference on taking trace of vector valued differential forms? Like if $A$ and$B$ are two vector valued forms then I want to understand how/why this equation is true? ...
1
vote
3answers
111 views

Construction of an osculating circle

Let $\alpha$ be a unit-speed curve.Then there exists a unique circle $\beta$ such that $\beta(0)=\alpha(0), \ \beta'(0)=\alpha'(0), \ \beta''(0)=\alpha''(0).$ Attempt: Consider $\beta(s)= \textbf{p} ...
1
vote
1answer
378 views

A difficult question about diffeomorphism about submanifold

Let $M$ and $N$ be two smooth manifolds, and $f: M \to N$ be a submersion , ${{f}^{-1}}(y)$ is compact for all $y$ in $N$. Then prove for any $x$ in $N$ there is an open neighborhood $U$ of $x$ such ...
1
vote
1answer
272 views

Covariant derivative of (1,1)-tensor

Suppose I have an endomorphism $J:TM \to TM$ and a connection on M. It is possible to define $\nabla_X J$ by transforming $J$ into a (1,1)-tensor and using the extension of $\nabla$ to tensors. Going ...
1
vote
1answer
119 views

Is the hypersurface of class $C^k$ a $C^k$-differentiable manifold?

In Folland's Introduction to Partial Differential Equations: A subset $S$ of ${\mathbb R}^n$ is called a hypersurface of class $C^k$($1\leq k\leq\infty$) if for every $x_0\in S$ there is an open ...
0
votes
4answers
312 views

Topologies and manifolds

This question might seem philosophical a bit: in a standard manifolds introductory course. when one talks about open , closed sets in $\mathbb{R}^n$ it's always the standard euclidian topology that ...
0
votes
6answers
576 views

Volume of spheres in higher dimensions?

What is the volume of spheres in higher dimensions?
10
votes
1answer
281 views

metric in the Wasserstein space of gaussian measures

I am reading the paper "Wasserstein Geometry of Gaussian measures" by Asuka Takatsu (section 3 is of interest to me) and I have difficulties understanding how the metric is used. In particular, I am ...
9
votes
1answer
305 views

uniqueness of the smooth structure on a manifold obtained by gluing

I've just read a proof that If $M$, $N$ are smooth manifolds with boundary and $f: \partial M\rightarrow \partial N$ is a diffeomorphism then $M \cup_f N$ has a smooth manifold structure such that ...
9
votes
1answer
145 views

When is a $k$-form a $(p, q)$-form?

Let $X$ be a complex manifold and denote the space of all $(p, q)$-forms on $X$ by $\mathcal{E}^{p,q}(X)$. Forgetting about the complex structure, we can consider the real differential $k$-forms on ...
8
votes
2answers
233 views

A curve whose image has positive measure

It is well-known that there are continuous curves $f:I \to \mathbb R^2$ (where $I \subset \mathbb R$ is an interval) whose image have positive measure (e.g Peano curve). I have read somewhere that if ...
7
votes
2answers
101 views

real meaning of divergence and its mathematical intuition

how does divergence which means sink or source equal to ∂Fx/∂x+∂Fy/∂y +∂Fz/∂z.I have been thinking it for a long time and i think "divergence tells us how fast the vector increases when we move apart ...
6
votes
1answer
248 views

Positive definiteness of Fubini-Study metric

Define the Fubini-Study metric $$g_{i\overline{j}} = \frac{\delta_{i\overline{j}}(1+|\boldsymbol{z}|^2)-\overline{z}^jz^i}{(1+|\boldsymbol{z}|^2)^2} $$ for $i,j=1,\ldots,n$ and $z_i$ complex variables ...