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
125 views

In which space does the Lie derivative of a vector field really live?

I'm slightly puzzled about the space in which the Lie derivatives of vector fields live. I get the worrying impression that it lives in the double tangent bundle, which I know is not true, so i need ...
1
vote
0answers
47 views

alternating bilinear form with wedge product

Let $\phi : \textbf{R}^4 \otimes \textbf{R}^4 \rightarrow \textbf{R}$ be an alternating bilinear form. Prove that there exist linear maps $\alpha, \beta :\textbf{R}^4 \otimes \textbf{R}^4 \rightarrow ...
1
vote
0answers
40 views

How is the Hessian defined under a different metric + definition?

I am reading on wikiedpia and the definition of Hessian $\mbox{Hess}(f)=\nabla_i\, \partial_j f \ dx^i \!\otimes\! dx^j = \left( \frac{\partial^2 f}{\partial x^i \partial x^j}-\Gamma_{ij}^k ...
1
vote
0answers
44 views

acoustic dipole volume integral w/ dirac delta?

I have an acoustic research problem that leads to the following integral formulation: \begin{align} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial ...
1
vote
0answers
48 views

a question about differential geometry

Let $S$ be a surface and $x: U\to S$ be a parametrization of $S$. If $ac-b^2 <0$, show that $$a(u,v)(\dot u)^2+2b(u,v)\dot u\dot v+c(u,v)(\dot v)^2=0$$ can be factored into two distinct equations, ...
1
vote
0answers
61 views

Can you have a “real” decomposition of differential forms on a symplectic manifold?

With a choice of (almost) complex structure on a symplectic (Kahler) manifold, has anyone thought of how or what it means to globally define "real" versions of the Dolbeault operators? i.e. Something ...
1
vote
0answers
102 views

Is a variety a CW-complex?

How to establish that any differentiable manifold and any complex algebraic variety is a CW-complex ? Thank you in advance for your help.
1
vote
0answers
67 views

Connection on Tangent Bundle of Group Manifolds

The thing puzzling me is about a transformation rule of connection on tangent bundle of group manifolds: Assume one has a compact and simply connected Lie group $K$. One can give a metric $g$ on $K$ ...
1
vote
0answers
30 views

Trouble proving identity - Gauge theory/Maurer-Carton one-form/Adjoint representation

The Identity I am trying to prove is the one in this already asked question how to show that ${ad}_{g_{\alpha\beta}} \circ g_{\alpha\beta}^{\star}\theta=-g_{\beta\alpha}^{\star}\theta$? The author ...
1
vote
0answers
99 views

A question about stereographic projection of a plane onto a sphere

I am reading a paper by Christine Bernardi, available here http://epubs.siam.org/doi/pdf/10.1137/0726068, my question relates to page 1237, which I shall elaborate on: In this part we have the unit ...
1
vote
0answers
66 views

Pappus theorem and area of a revolution surface.

Let $y=f(z)$ be a function. How can I calculate the area of the surface obtained rotating the function along the $z$ axis, where y is the revolution torus? Is it possible to do it using Pappus formula ...
1
vote
0answers
33 views

Construction of a diffeomorphism handling varying domain

Let $\Omega$ be a strictly convex domain, $\partial\Omega\in C^{2,1}$ We define a foliation $\{\Omega_t\}_{0\leq t\leq 1}\subset\Omega$ as follows. Let $\Omega_0=B_r(x_0)$, a small ball centered at ...
1
vote
0answers
24 views

Normal Curvature $κ_n(p, v)$

Let $S$ be the embedded torus with parametrization $σ(θ, ϕ) = ((2 + \cos θ) \cos ϕ,(2 + \cos θ) \sin ϕ,\sin θ)$. The first and the second fundamental forms of $σ$ are $dθ^2 + (2 + \cos θ)^ 2 dϕ^2$ ...
1
vote
0answers
38 views

Why the following is not a tensor?

Say we have an arbitrary coordinate system, in which a position vector is represented by: $$\vec V=Z^i\vec Z_i$$ Where $\vec Z_i$ is a covariant basis. Now $\partial {\vec Z_i}/\partial Z^j$ term has ...
1
vote
0answers
25 views

integration of differences of two inverse functions of CDF

When F, G are both cumulative distribution functions, $\int_0^1 | F^{-1}(y) - G^{-1}(y)|dy = \int_{-\infty}^\infty |F(x) - G(x)|dx.$ The paper (Vallender 74') says that "The equation follows from ...
1
vote
0answers
55 views

References about algebraic/differential geometry in French

Aside from learning mathematics, I am learning French, so I would like to practise both at the same time if possible. Do you know of any good references about complex/algebraic geometry or ...
1
vote
0answers
45 views

Why are free objects related to tensors important?

Let $R$ be a commutative ring and $M$ be an $R$-module. Then, we can construct the tensor algebra $T(M):=\oplus_{n\in \mathbb{N}} M^{\otimes n}$ which is likely a free object. That is, for any ...
1
vote
0answers
36 views

Fibered product of symplectic fiber bundles

Let $P_{1}\to B$ and $P_{2}\to B$ be smooth fiber bundles over the same base manifold $B$, and let us assume that the total spaces $(P_{1},\omega_{1})$ and $(P_{2},\omega_{2})$ are symplectic ...
1
vote
0answers
18 views

Parametric equation: tangents

I'm not sure what it's called in English, but a curve can have different types of tangents and if a parametrization of the curve is given by $r(t)$, then if $r'(a) = 0$ and $r''(a) \neq 0$, then at ...
1
vote
0answers
30 views

Differential Geometry in 3D vs Differential Geometry in n dimensions

I have read (in the introduction to Elementary Differential Geometry by Pressley) that some theorems in 3D differential geometry cannot be extended to differential geometry in n dimensions. Is there a ...
1
vote
0answers
33 views

Quick question about curves and basis

Hello all I have a quick question because I am trying to understand my notes and I am confused. Can anyone here atleast give me a hint or anything! Taking note of the fact that the normal vector, ...
1
vote
0answers
81 views

Isothermal parameterization, Inverse of the Gauss Map

This problem is from Do Carmo's Differential Geometry of Curves and Surfaces. It is question 13 from chapter 3.5, to be specific. Suppose that S is a minimal surface without any umbilical points ...
1
vote
0answers
82 views

Vector field tangent to a proper submanifold

Let $S\subset M$ be a properly embedded submanifold (in particular, $S\subset M$ is a closed submanifold) and let $V$ be a smooth vector field on $M$ tangent to $S$. Since we have that $M\setminus ...
1
vote
0answers
66 views

Lagrangian Method for Christoffel Symbol and (non-)holonomic basis

I rencently learned about the lagrangian/variational method for computing Christoffel symbols. Let $\mathcal{M}$ be a $m-$dimensional manifold with $g_{ij}$ being the metric tensor components and ...
1
vote
0answers
79 views

Calculation of Poincare dual to $\mathbb{CP}^n$ in $\mathbb{CP}^{n+1}$

I would like to go about finding an explicit representative of the Poincare dual to $\mathbb{CP}^n$ in $\mathbb{CP^{n+1}}$, I am following Bott & Tu and would like an explicit form to be wedged ...
1
vote
0answers
73 views

Comparing normal bundles of embedded submanifolds and their sections.

Let $M,M'\subseteq \mathbb{R^n}$ two compact embedded submanifolds, which are abstractly diffeomorphic. Tangent and normal bundle of the two submanifolds inherit a metric from $\mathbb{R^n}$. By the ...
1
vote
0answers
99 views

What are these integrals used for?

In a primer on differential geometry, it was mentioned that each of these are different types of line integrals: $\int_C f(\vec r) |d\vec r|$, $\int_C \vec F |d\vec r|$, $\int_C f(\vec r)d\vec r$, and ...
1
vote
0answers
129 views

Complex Manifold: Stokes

This is a lemma for: Helffer-Sjöstrand Given the complex plane. Consider a smooth function: $$f_E\in\mathcal{C}^\infty_c(\mathbb{R}^2):\quad\bar{\partial}f_E\restriction_\mathbb{R}=0$$ How to apply ...
1
vote
0answers
57 views

Topology and Smooth Structure on the Bundle of Covariant $k$-Tensors

(All vector spaces are finite dimensional and real). Given a vector space $V$, let $T^k(V^*)$ denote the vector space of all the covariant $k$ tensors on $V$. Following Lee's Introduction to Smooth ...
1
vote
0answers
237 views

$F$-related vector fields

I'm having a bit of difficulty understanding $F$-related vector fields. I think I understand conceptually what's going on (taking a vector field on a manifold $M$ and getting a smooth vector field on ...
1
vote
0answers
26 views

Dimension of scalar solutions to these self-dual/anti-self-dual equations

Let $M$ be a 4-dimensional Riemannian manifold. Let $\kappa$ be a 1-form. I look for solution function $\phi$, such that there exists functions $\alpha$ and $\beta$ \begin{equation} {\left( ...
1
vote
0answers
50 views

$SO(3)$ has a subgroup $U(1) \times U(1)$?

I am wondering - and asking you - whether there is a subgroup $U(1) \times U(1)$ of the Lie group $SO(3)$. Equivalently, I can reformulate it from a geometrical point of view: does there exist a torus ...
1
vote
0answers
47 views

Problems in understanding the notion of “active” diffeomorphisms on a manifold

I have been studying differential geometry recently in the hope of gaining a deeper understanding of General Relativity (GR). I have run into an issue when trying to understand the notion of what is ...
1
vote
0answers
226 views

How to evaluate solid angle subtended by an ellipse at any arbitrary point on the vertical axis passing through the center

`Question: How to evaluate the exact value of solid angle subtended by an ellipse (or elliptical plane) at any arbitrary point lying on the vertical axis passing through the center. Standard equation ...
1
vote
0answers
59 views

the gradient in complex coordinates

Let $M$ be an $n$-dimensional complex manifold, or equivalently a $2n$ real manifold. Let $g$ be a Riemannian metric. Let $f\in C^\infty(M)$. What is $\nabla f$? If $x_1,\dots,x_n,y_1,\dots,y_n$ are ...
1
vote
0answers
80 views

Relative version of Whitney embedding's theorem (reference needed)

One form of Whitney embedding theorem says that if $M,N$ are smooth compact manifolds and the dimension of $N$ is more than twice the dimension of $M$, then the space of embeddings $M\to N$ is open ...
1
vote
0answers
39 views

Uniform convergence of the harmonic form heat flow

[${\bf NOTATIONS}$] Let $M$ be a closed Riemannian manifold of $m$ dimensional, $p\in\{1,\cdots,m\}$. $A^p:=\{\text{smooth p-forms on }M\}$. $\delta:A^{p+1}\to A^p$ denotes the formally adjoint ...
1
vote
0answers
61 views

Integration on $ \mathbb{P}^n ( \mathbb{R} ) $.

Could you tell me please, when and how we calculate the integral of a function on $ \mathbb{P}^n ( \mathbb{R} ) $ ? Do you have some references about that ? Thank you in advance.
1
vote
0answers
73 views

confusion about basic Kahler geometry

I am really struggling to understand the basics of Kahler geometry and hope someone can give me some guidance. Suppose we have a complex manifold with some complex structure $J$ and let $g$ be a ...
1
vote
0answers
83 views

Coding for a calculation in differential geometry using Maple

I am beginner in maple. And my field is Differential geometry. I've learnt lie brackets using maple help. But I am testing this calculation through maple. I have these vector fields $e1=z^2\ast ...
1
vote
0answers
79 views

Induced Lie Algebra Representation, Left invariant vector fields and more…

The following is an excerpt from a proof in John Lee's Introduction to Smooth Manifolds I am struggling to understand. I would appreciate if someone was able to help me with whatever it is I am ...
1
vote
0answers
80 views

Direction of outward normal differential $\hat n\ ds$

In vector calculus, why is the outward normal differential $\hat n\ ds$ around a closed curve $dy\ \hat i - dx\ \hat j$? Why isn't it $-dy\ \hat i + dx\ \hat j$ or something else?
1
vote
0answers
36 views

determinant of metric on complexified vector space

Let $M$ be a complex manifold of dimension $n$ with Hermitian metric $g$. Extend $g$ to $TM^{\mathbb{C}}$ linearly. I believe that at each $p$ we can say a basis for $T_pM$ as a real vector space is ...
1
vote
0answers
106 views

Closed 1-forms on Simply Connected Manifold

Is it true that closed 1-forms on a simply connected differentiable manifold are exact. If so, could you explain why? Thanks very much
1
vote
0answers
22 views

When is the composition of multiple flows the identity?

Suppose I have some number of vector fields $a,b,\ldots,k$, and I denote flow along $a$ for time $t$ by $a_t$, etc. When is it the case that for all points $x$, $a_tb_tc_t\ldots k_tx = x$? In the ...
1
vote
0answers
52 views

Left invariant vector field question

This question is an exercise from Jeff Lee's Manifolds and Differential Geometry Let $\tilde{X}$ be the left invariant vector field corresponding to $X\in L(V,V)$ (That is, $\tilde{X}_g=(dL_g)_e X$). ...
1
vote
0answers
41 views

Minimising surface with given curve as a boundary

I have a problem connected to finding a minimal surface with a given boundary. I know that it is the surface with zero mean curvature but as I have to obain differential equations for such surface I ...
1
vote
0answers
28 views

Plane fields transversal to 1-dimensional bundles and Ehresmann connection

This is a question related to something I saw in the book Confoliations by Thurston and Eliashberg. Consider three G-bundles $M \rightarrow F$ and two dimensional plane fields $\xi$: A. $M= F\times ...
1
vote
0answers
146 views

Integration over a second order tensor

I would like to compute the mean value of a second order tensor $\mathbf{T}$ expressed in planar cylindrical coordinates. The mean value for any second order tensor is (reference [1] page 101) ...
1
vote
0answers
38 views

A connection is uniquely determined by its action on global sections

Let $\mathcal{A}^i(E)$ be the sheaf of $C^{\infty}$ differential $i-$form with values in $E$, where $\pi: E \mapsto M$ is a complex vector bundle. We define connection on $E$ a $\mathbb{C}-$linear ...