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
131 views

Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$. In this question, it is proven that the canonical map ...
4
votes
1answer
97 views

what does it mean for a differential form to be well defined on a manifold?

What does it mean for a differential form to be well defined on some manifold. In particular, why the $2$-form $\omega=d\psi\wedge d\theta$ is well defined on $S^{2}$? Thank you in advance.
3
votes
1answer
110 views

Normal bundle of the two-dimensional sphere manifold embedded in $\mathbb R^4$

Let $M \subset \mathbb R^4$ be a smooth manifold diffeomorphic to $S^2$. How can one prove that normal bundle of $M$ has at least one non-vanishing global section. I think that $M$ should be ...
1
vote
0answers
53 views

integral of a 2-form over an oriented manifold

An old exam question: Let $M$ be oriented submanifold of $\mathbb{R}^{n}$ of dimension $k$, let $\omega$ be a $k$-form on $M$. 1) Define $\displaystyle\int_{M}\omega$ 2) let ...
3
votes
0answers
94 views
+50

A singular $n-$cube and a circumference defined the border than 2-cube

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
2
votes
1answer
131 views

De Rham cohomology for $\mathbb{R^2}$

De Rham cohomology groups for $\mathbb{R^2}$. $H^{0}_{dR}(\mathbb{R}^{2})=\mathbb{R}$ since $Z^{0}(\mathbb{R}^{2})$ is the one dimensional space of locally constant functions on $\mathbb{R}^{2}$ and ...
6
votes
1answer
407 views

Relations of Characteristic classes: Chern, Stiefel-Whitney, Pontryagin, Euler, Wu class. [closed]

We have a quite some characteristic classes: Chern class, Stiefel-Whitney class, Pontryagin class, Euler class, Wu class, etc. I wonder whether some math experts can use simple words and basic ...
7
votes
2answers
864 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 ...
4
votes
3answers
457 views

Surface with constant Gaussian curvature $K > 0$

Besides the sphere, is there any other surface with constant and positive Gaussian Curvature $K$?
1
vote
1answer
56 views

projective homogeneous $G$-variety is equivariantly isomorphic to a partial flag variety $G/P$ where $P$ is projective variety.

I am looking for a proof(or refference) for this fact A projective homogeneous $G$-variety is equivariantly isomorphic to a partial flag variety $G/P$ where $P$ is parabolic subgroup.
3
votes
1answer
372 views

volume of projective space $\text{Vol}(\mathbb CP^N)$

How can we compute the volume of projective space $$\text{Vol}(\mathbb CP^N)$$
2
votes
1answer
204 views

curve integral - intersection between plane and sphere

I am going to calculate the line integral $$ \int_\gamma z^4dx+x^2dy+y^8dz,$$ where $\gamma$ is the intersection of the plane $y+z=1$ with the sphere $x^2+y^2+z^2=1$, $x \geq 0$, with the orientation ...
1
vote
1answer
137 views

Can I construct an affine connection on a Riemannian manifold from arbitrary Christoffel Symbols?

The question is rather simple. All my definitions are as in Do Carmo's "Riemannian Geometry". If $M$ is a Riemannian Manifold, can I construct an affine connection $\nabla$ on it by setting, for all ...
1
vote
0answers
31 views

Minimization Problem for Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
0
votes
0answers
90 views

Calculate the geodesic of Z = XY between two points

I have only learned about calculus and linear algebra, so I don't know about differential algebra. I got to know about the concept of "geodesic" recently. What I need to know is this: Suppose I ...
2
votes
1answer
108 views

Connection vs Curvature

Why is twice a connection usually referred to the curvature: $\overline{\nabla}\circ\nabla=F^\nabla$ Is there an axiomatic definition of curvature, e.g. it is module-linear operator etc?
2
votes
1answer
49 views

Under what conditions the kernel and image of a linear bundle map are subbundles?

Under what conditions the kernel and image of a linear bundle map are subbundles?
4
votes
1answer
119 views

The diffential of commutator map in a Lie group

Leb $G$ be a Lie group and $f:G\times G\rightarrow G$ be the commutator map $:(x,y)\mapsto xyx^{-1}y^{-1}$. How to obtain the Lie bracket in the associated Lie algebra of $G$ from the derivatives of ...
2
votes
1answer
72 views

Computing the volume element of an oriented Riemannian manifold

I'm reading Gallot-Hulin-Lafontaine, and in section 2.7 they say they following: I wanted to check that the second $v_g,$ given in a local oriented chart, satisfied the first property. So I ...
0
votes
0answers
48 views

Ricci curvature tensor calculation?

Is this correct? $$ R=R_{ab}g^{ab} $$ $$ R(g_{cd})=R_{ab}g^{ab}(g_{cd}) $$ $$Rg_{cd}=R_{ab}g^{ab}(g_{ab}\delta^a_c\delta^b_d)$$ $$Rg_{cd}=R_{ab}(g^{ab}g_{ab})\delta^a_c\delta^b_d$$ ...
1
vote
1answer
67 views

Kernel Of surrjective linear bundle map?

$E$ is a vector bundle on $M$. let $\phi:E\longrightarrow TM$ be a surjective linear bundle map. Is $\ker\phi$ vector sub-bundle of $E$?
1
vote
0answers
41 views

Motivation for tensor density

Wiki has provided the basic definitions of the tensor density, but what I really want to know is the motivation and the advantage of this concept. Could anyone give me some ideas?
0
votes
1answer
92 views

Stokes Theorem: Unit Ball

Given the unit ball: $$M:=\mathbb{B}:\quad\partial M=\varnothing$$ Consider the top-degree form: $$\omega:=1\mathrm{d}x\wedge\mathrm{d}y=\mathrm{d}(x\mathrm{d}y)=:\mathrm{d}\Omega$$ Then one has by ...
0
votes
2answers
203 views

Homotopy invariance of line integral on manifolds

Consider a 1-form: $\omega\in\Gamma(\mathrm{T}^*M)$ and two differentiable curves: $\gamma,\tilde{\gamma}:[a,b]\to M:\gamma(a)=\tilde{\gamma}(a),\gamma(b)=\tilde{\gamma}(b)$ together with a ...
2
votes
1answer
142 views

Moment map of the action of $SU(2)$ on $\mathbb C^{2n}$

Let $SU(2)$ acts on symplectic space $((\mathbb C^2 -\ (0,0))^{n},\omega)$, where $$\omega=dx_1\wedge dx_2+dx_3\wedge dx_4+\cdots+dx_{4n-3}\wedge dx_{4n-2}+dx_{4n-1}\wedge dx_{4n}$$ as ...
0
votes
1answer
147 views

Derivative of riemannian metric

I dont understand the following detail $$ \frac{1}{2} \int_a^b \frac{d}{dt}(g(X,X)ds = \int_a^bg(\nabla_YX, X)$$ Here $X = d\phi (\partial/\partial s)$ and $Y =d\phi (\partial/\partial t)$. Where ...
3
votes
2answers
268 views

Ricci Soliton geometric meaning

I wonder what is the geometrical, intuitive meaning of a Ricci soliton on a manifold. The definition that I use is as follows. $V$ is a vector field on the manifold, $g$ is a Riemannian metric. ...
1
vote
0answers
65 views

Commutative differential graded algebra

The situation is the following: Let $V$ be a Lie algebra of finite dimension, say n, and let be the graded commutative algebra $(\bigwedge^{\bullet}V)^*:=(\bigoplus_{k=0}^n\bigwedge^kV)^*$ and define ...
1
vote
1answer
77 views

number of points of tangency of the zero divergence vector field and the equator of the sphere.

Let $V$ be vector field on the sphere $S^2$ and $\operatorname{div} V=0$. What is the minimum number tangency points of this vector field and the equator of the sphere?
1
vote
1answer
108 views

Integrating a 0-form

The Stokes theorem states: $$\int_\mathcal M d\omega =\int_{\partial \mathcal M} \omega $$ If we have that $\mathcal M$ is a one dimensional manifold with two extreme points, like a closed interval ...
0
votes
1answer
44 views

Unicity solution in this differential equation

I'm studying Sherk surface which is the unique minimal surface with parametrization given by $\phi(x,y)=(x,y,f(x)+g(y))$. Using the mean curvature formula, is easy to show that this surface is minimal ...
0
votes
2answers
78 views

How to find the equation of the normal line to the surface S

How to find the equation of the normal line to the surface $S$: $$f(u,v)=(2u-v,u^2+v^2,u^3-v^3)$$ at the point $M(3,5,7)$? Could someone post the complete solution?
4
votes
2answers
894 views

Definition of “a topological manifold with corners”.

How can we define a topological manifold with corners and its corners? Then, do we use "invariance of domain" to define corners, as we really need this theorem in order to define "boundaries of a ...
2
votes
1answer
111 views

Question about the second fundamental form

I am studying Riemannian geometry and have a question understanding something. I use Do Carmo's book. In the book, a vector field is defined for isometric immersions: for an immersion $$ ...
0
votes
1answer
62 views

Cocycles vector bundles and metrics

It is well known, and not difficult to prove that a vector bundle $E$ over a (smooth) manifold $M$ together with a metric gives rise to orthonormal frames (by Gram-Schmidt). An consequnece is that the ...
1
vote
2answers
416 views

Radius of curvature for the plane curve $x^3 + y^3 = 12xy$.

Could someone help me with this problem? : Determine the radius of curvature for the plane curve $x^3 + y^3 = 12xy$ at the point $(0, 0)$.
2
votes
1answer
192 views

Reference request-What is the prerequisite of S.S.Chern's proof of the generalised Gauss-Bonnet theorem?

The title basically explains everything. The OP is an independent learner, who in the current stage sets S.S.Chern's proof of the generalised Gauss-Bonnet theorem as the goal. But what is the ...
1
vote
2answers
677 views

Area of a weird ellipse shape.

A propeller has the shape shown below. The boundary of the internal hole is given by $r = a + b\cos(4α)$ where $a > b >0$. The external boundary of the propeller is given by $r = c + d\cos(3α)$ ...
1
vote
2answers
77 views

Why do we need three indices for Christoffel Symbols

I read the following results on covariant differentiation (summation convention applied): If $X=X^i e_i$ and $Y=Y^je_j$, then $$\nabla_XY = X^i\nabla_{e_i}(Y^je_j) = X^ie_i(Y^j)e_j + ...
2
votes
1answer
127 views

Intuition/visualization for a non-flat connection

I'd just like to check whether my visualization for a way to get a non-flat connection is correct. The definition I am using for a connection is, for a fiber bundle $\rho:E \to B$, a smooth ...
0
votes
1answer
31 views

Why this doesn't transform properly?

We are in $\mathbb R^n $, with a tensor field of components $T_\nu$, and being $e_\mu$ the vectors of the basis: $e_\mu \equiv \partial_\mu$, then I'm asked to show that $\partial_\mu T_\nu$ can't ...
2
votes
2answers
329 views

Recover Covariant Derivative from Parallel Transport

It is well known that one can recover the connection from the parallel transport. I struggle to understand this concept. Since $\Gamma(\gamma)^t_s:E_{\gamma(s)}\to E_{\gamma(t)}$ is an isomorphism ...
10
votes
1answer
216 views

Soft question: How does basic differential geometry “fit together”?

I'm self-studying diff geom from Lee's Introduction to Smooth Manifolds. He warns the reader that there's a lot of machinery to construct, which is fine, and he explains things with wonderful clarity. ...
1
vote
1answer
42 views

Find the derived of an implicit given function.

Let $C=\{(x,y,z) \in \mathbb(R)^3| \sin x + \sin^2 y + \sin^4 z=0 \ \text{and} \ (x-z)^2=4\pi^2\}.$ By the implicit function theorem, we have that $C$ can be parametrized as a smooth curve in the ...
4
votes
0answers
117 views

Tangent developable of helix.

Let $T$ be union of tangent lines to helix $C=(\cos x, \sin x,x)$. 1) I want to prove that $T - C$ is a smooth manifold and find equation for $T$. 2) I want to find how many times a line can ...
6
votes
2answers
193 views

How many differential forms on the complex plane?

I am puzzled by the fact that the two differential forms $$\begin{array}{cc} dz=dx+i dy , & d\overline{z}=dx-i dy \end{array} $$ are $\mathbb{C}$-linearly independent, even if the underlying ...
3
votes
3answers
121 views

Why is $d*F$ equal to $\partial _\mu F^{\mu \nu}$?

Given that $A = A_\nu dx^\nu$ and $F = \partial_{\mu}A_\nu dx^\mu \wedge dx^\nu$ Why does $d*F$ equal to $\partial _\mu F^{\mu \nu}$? How does all the $\frac{1}{2}\varepsilon^{abcd}F_{cd}$ fit into ...
-2
votes
1answer
53 views

reduced space of coadjoint orbit

Let $G$ be a compact Lie group and $\lambda\in \frak{g}^*$$=(Lie G)^*$ and $O_\lambda$ be the Coadjoint orbit through $\lambda\in \frak{g}^*$ and $\mu:O_\lambda\to\frak{g}^*$ be the moment map, ...
0
votes
1answer
17 views

mapping between differential forms and its property

I am trying to prove the following property of the map between differential forms: (Spivak's book ''Calculus on manifolds'' p.91) $$f^{\star}\;\Lambda^{k}(\mathbb{R}^{m}_{f(p)})\to ...
0
votes
1answer
36 views

Vectors that span normal bundle of a submanifold

If $E_{1}=\frac{\partial}{\partial y}-kx\frac{\partial}{\partial w}$ and $E_{2}=cos\psi\frac{\partial}{\partial x} + sin \psi\frac{\partial}{\partial w}$ span tangent space of $M$, where $M$ is a ...