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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90 views

Relation between two Riemannain connections

Let $g$ be a Riemannian metric on $M$ and let $\tilde{g}=f^{2}g$ where $f$ is a smooth function that is never zero. let $\nabla$ and $\nabla'$ be the Riemannain connections of $g$ and $\tilde{g}$ on ...
4
votes
4answers
567 views

help in understanding tangent vectors

In Aaron's answer here... "Given a manifold $M$, and a point $p\in M$, we have a vector space $T_pM$ of the tangent vectors to $M$ at $p$. For example, if you take the hollow sphere sitting inside ...
13
votes
1answer
556 views

Tautological 1-form on the cotangent bundle

I'm trying to understand a little bit about symplectic geometry, in particular the tautological 1-form on the cotangent bundle. I'm following Ana Canas Da Silva's notes. On page 10 she describes the ...
13
votes
3answers
575 views

Where do we need the axiom of choice in Riemannian geometry?

A friend of mine is a differential geometer, and keeps insisting that he doesn't need the axiom of choice for the things he does. I'm fairly certain that's not true, though I haven't dug into the ...
12
votes
1answer
401 views

Local triviality of principal bundles

Suppose I define a principal $G$-bundle as a map $\pi: P \to M$ with a smooth right action of $G$ on $P$ that acts freely and transitively on the fibers of $\pi$. Does it follow that $P$ is locally ...
10
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1answer
241 views

Infinite dimensional constant rank theorem

Suppose you have an analytic map $\phi : E \rightarrow \mathbb{C}^n$, where $E$ is a complex Banach space, and such that the rank of $D \phi$ is constant. Is it true then that the set ...
10
votes
1answer
524 views

Is there a Stokes theorem for covariant derivatives?

A $V$-valued differential form on $M$ is a smooth map $\omega : TM \to V$ such that $\omega$ restricted to any tangent space $T_p M$ is an element of the $V$-valued exterior algebra $\Lambda^n (T_p M, ...
10
votes
2answers
676 views

What is the relation between connections on principal bundles and connections on vector bundles?

I'm reading Kobayashi / Nomizu 's vol. I. I am reading about connections in principal G-bundles. After that chapter there's a chapter on (linear) connections on Vector bundles. Since we can associate ...
9
votes
1answer
163 views

Prove the curvature of a level set equals divergence of the normalized gradient

Suppose we have a function $\phi : \mathbb{R}^2 \to \mathbb{R}$, and a curve $\gamma:\mathbb{R}\to\mathbb{R}^2$ defined by a level set of $\phi$, ie. the codomain of $\gamma$ is ...
9
votes
1answer
187 views

Which coefficients of the characteristic polynomial of the shape operator are isometric invariants?

Let $M^n \subset \mathbb{R}^{n+1}$ be an isometrically immersed Riemannian hypersurface. The shape operator $s$ is the $(1,1)$ tensor field characterized by $$\langle X, sY \rangle = \langle ...
8
votes
1answer
762 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 ...
8
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1answer
2k views

Intuitive explanation of Left invariant Vector Field

Intuitively what is meant by a left invariant vector field on a manifold?
7
votes
2answers
172 views

Interpretation of $p$-forms

Let $M$ be a smooth manifold, let $C^{\infty}(M)$ be set of all smooth functions from $M$ to $\mathbb R$ and let $Vec(M)$ denote the set of all vector fields on $M$. A $1$-form on $M$ is a ...
7
votes
1answer
140 views

Uniqueness of curve of minimal length in a closed $X\subset \mathbb R^2$

Suppose $X$ is a simply connected closed subset of $\mathbb R^2$. Let $a,b$ belong to $X$. Is it true that there is at most one curve inside $X$ from $a$ to $b$ such that the length of the curve is ...
7
votes
2answers
160 views

Definition of the principal symbol of a differential operator on a real vector bundle.

I'm trying to understand the construction of the dirac operator on a manifold, but actually I guess that doesn't really matter for the question at stake. I'm interested in understanding a definition ...
7
votes
1answer
225 views

Is the Structure Group of a Fibre Bundle Well-Defined?

Am I right in thinking that the structure group of a fibre bundle is any group $G$ of homeomorphisms of the fibre $F$ such that all transition functions map into $G$? Or is $G$ somehow the minimal ...
6
votes
1answer
150 views

Which textbook of differential geometry will introduce conformal transformation?

Which textbook of differerntial geometry will have these formulas about conformal transformation? $$\tilde g_{ij} = e^{2\varphi}g_{ij}$$ $$\tilde \Gamma^k{}_{ij} = \Gamma^k{}_{ij}+ ...
6
votes
3answers
502 views

Differential Forms and Vector Fields correspondence

Barrett O'Neill's Differential Geometry book says that Classical vector analysis avoids the use of differential forms on $\mathbb{R}^3$ by converting 1-forms and 2-forms into vector fields via the ...
6
votes
1answer
155 views

Possibilities of an action of $S^1$ on a disk.

I'm dealing with actions of the circle over differentiable manifolds. In the book I'm reading, they use the fact that an action of $S^1$ over a disk has to be equivalent (there has to exist an ...
6
votes
4answers
484 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 ...
6
votes
1answer
285 views

Are these definitions of submanifold equivalent?

Let $M$ be a manifold – by which I mean a second countable Hausdorff smooth manifold. Here's an "obviously" correct definition of (embedded) submanifold: Definition A. A subspace $S$ of $M$ is a ...
6
votes
1answer
280 views

An example of a derivation at a point on a $C^k$-manifold which is not a tangent vector

Let $M$ be a real $C^k$-manifold, $\mathscr{H}_x$ be the $\mathbb{R}$-algebra of germs of $C^k$-differentiable functions at $x \in M$. It is easy to show that $T_x M$ can be embedded into the space of ...
5
votes
2answers
374 views

Spin manifold and the second Stiefel-Whitney class

We know that: Spin structures will exist if and only if the second Stiefel-Whitney class $w_2(M)\in H^2(M,\mathbb Z/2)$ of $M$ vanishes. Can someone use simple words and logic to show why the ...
5
votes
1answer
75 views

integration in five dimensions space

I am doing this problem: Consider the differential form $$a=p_1 \, dq_1+p_2 \, dq_2-(p_1^2+p_2^2+q_1^2+q_2^2) \, dt\text{ in }\mathbb R^5=(p_1,p_2,q_1,q_2,t).$$ (a) Compute the differential $da$ and ...
5
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1answer
719 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
3answers
361 views

$\Omega=x\;dy \wedge dz+y\;dz\wedge dx+z\;dx \wedge dy$ is never zero when restricted to $\mathbb{S^2}$

We define a 2-form $\Omega$ on $\mathbb{R^3}$ by $\Omega=x\;dy \wedge dz+y\;dz\wedge dx+z\;dx \wedge dy$. How can I show that $\Omega|_\mathbb{S^2}$ is nowhere zero? Before proving that how can I ...
4
votes
2answers
207 views

Prove an identity about $\iint_S\mathbf{r}\wedge d\mathbf{S}$ using Stokes' theorem

$$ \int_C\mathbf{r}(\mathbf{r}\cdot d\mathbf{r})=\iint_S\mathbf{r}\wedge d\mathbf{S} $$ With $\mathbf{r} = (x,y,z)$ being a 3-dimensional vector. How do you get this result using Stokes' theorem?
3
votes
1answer
256 views

Integral of a curve with respect to its curvature?

I've been struggling with this one for about $3$ weeks: What is the integral of a $\mathbb{R}^3$ curve with respect to its curvature? I though about approaching it with the Ferret-S formulas, and ...
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0answers
102 views

Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...
1
vote
1answer
1k views

Lie derivative of a vector field equals the lie bracket

Let $X$ and $Y$ be vector fields on a smooth manifold $M$, and let $\phi_t$ be the flow of $X$, i.e. $\frac{d}{dt} \phi_t(p) = X_p$. I am trying to prove the following formula: $\frac{d}{dt} ...
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votes
1answer
299 views

Another vector bundles over of a Riemannian manifold

How is that any other vector bundle over a Riemannian manifold can be turn into a Riemannian manifold as well? I think that the question is of interest due to the presence in math.SE of several ...
14
votes
2answers
469 views

Is the set of singular matrices ever a differentiable manifold?

I can see that invertible matrices are a differentiable manifold however I don't know how to show that something is not a differentiable manifold so easily. Is it ever the case that singular matrices ...
13
votes
0answers
733 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
1answer
280 views

The Quaternions and $SO(4)$

I am interested in the map $\phi:S^3 \times S^3 \to GL_4(\mathbb{R})$ given as follows: Let $(p,q) \in S^3 \times S^3$. We identify $p$ and $q$ as real quaternions with unit norms and define ...
11
votes
2answers
631 views

How to introduce stress tensor on manifolds?

I want to understand the type of stress tensor $\mathbf{P}$ in classical physics. Usually in physics it is said that the force $\text d \boldsymbol F$ (vector) acting on an infinitesimal area $\text ...
10
votes
1answer
125 views

Is every compact hypersurface contained in a sphere which it touches twice?

Let $M\subset \mathbb{R}^{n+1}$ be a compact $n$-manifold. There exists, then, a smallest $n$-sphere containing $M$, and it must touch it in one point. Must it touch it twice? This seems quite ...
10
votes
2answers
258 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
3answers
515 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 ...
8
votes
1answer
145 views

Almost complex structure on $\mathbb CP^2 \mathbin\# \mathbb CP^2$

Why doesn't $\mathbb CP^2 \mathbin\# \mathbb CP^2$ admit an almost complex structure?
7
votes
2answers
110 views

Chern-Weil: why do we divide by $2\pi$?

So here's a somewhat incoherent question. To define characteristic classes in the Chern–Weil way, one takes a curvature form $\Omega$ on a vector bundle $E \to M$ and an invariant polynomial ...
7
votes
2answers
785 views

how to compute the de Rham cohomology of the punctured plane just by Calculus?

I have a classmate learning algebra.He ask me how to compute the de Rham cohomology of the punctured plane $\mathbb{R}^2\setminus\{0\}$ by an elementary way,without homotopy type,without ...
7
votes
1answer
437 views

Continuous maps between compact manifolds are homotopic to smooth ones

If $M_1$ and $M_2$ are compact connected manifolds of dimension $n$, and $f$ is a continuous map from $M_1$ to $M_2$, f is homotopic to a smooth map from $M_1$ to $M_2$. Seems to be fairly basic, ...
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votes
2answers
522 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
173 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
1answer
393 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
322 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 ...
5
votes
1answer
108 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
191 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
3answers
175 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 ...
5
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0answers
370 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 ...