Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

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If $\omega$ is a differential $4$-form on a $10$-manifold $M$ then $\omega \wedge d\omega$ is exact

Let $\omega$ be a differential $4$-form on a $10$-manifold $M$. I am trying to show that $\omega \wedge d\omega $, which is a $9$-form, is exact. Clearly $\omega \wedge d\omega$ is closed, because $...
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When is a functor the associated bundle construction?

Let $G$ be a group and $X$ be a space. I am principally interested in these objects in the category of schemes over some base $S$, but an answer in another geometric category would be welcome. ...
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Definition of Symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. I'm trying to see why the non-degeneracy of $\omega$ is equivalent to say that the contraction map $X \longmapsto i_{X}\omega$ defines an isomorphism between ...
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Is regularity of surfaces a diffeomorphism invariant?

Knowing the definition of regular surface (Definition of a regular surface). Let S be a diffeomorphic surface to $A\subset\mathbb{R}^3$. Is $A$ a regular surface as well?
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Curve. Find a parameterization and critical points

The curve (c): $ x^2+y^2=z^2 $ \ $ \ln z=\operatorname{arctg}\frac{y}{x}, x\neq0, z>0 $ Find a parametrization, the critical points, a natural parametrization and the curvature and torsion at ...
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A question about effective smooth action

Let $G$ be a Lie group acting smooth and effectively in a manifold $S$. Let $M$ be another manifold and $f:M \longrightarrow G$ a function such that the function: $$ F:M\times S \longrightarrow S$$ $$ ...
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1answer
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Homogeneous fns

A function $g$:$\mathbb{R}^3→\mathbb{R}$ is said to be homogeneous of degree $k$ if $g(tx,ty,tz)=t^kg(x,y,z)$, $t>0$. If $g$ is differentiable and homogeneous of degree $k$, then $$xg_x+yg_y+zg_z=...
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What kind of manifold doesn't have a specific rank distribution?

For example, I know that $2$ dimensional sphere doesn't have rank $1$ tangent distribution for it vanishes at some point by the hairy ball theorem. Is there any criterion for it?
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Frenet frame for curve

Find the Frenet frame for the curve $C: z=x^2+y^2, x=y$ at the point $(1,1,2)$. Can somebody help me with this problem? I tried some ideeas and I didn't manage to find an answer. This is the first ...
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Pursuit-evasion game with $n$ pursuers and $1$ evader

Assume $n$ pursuers ($P_i$) at the vertices of an $n$ sided regular polygon with the evader ($E$) at the centre. For what all $n$ can be the evader be caught? Pursuers and evader have same speed ...
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Well-definedness of the operator $\partial$

Let $M$ be a Riemann surface. Define the operator $\partial\colon\Omega^0 (M)\sqcup \Omega^1 (M)\to \Omega^1 (M)\sqcup \Omega^2 (M)$ as \begin{equation} \partial f = f_z dz,\ \partial\omega = v_z dz\...
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Proving postulate about a property fo spherical vectors

Assume we have $X, Y$ constant unit vectors of $\mathbb{R}^3$ I postulate that the maximum of the function: $(V \cdot X) (V \cdot Y)$ I reached by the halfway vector between $X,Y$ i.e at the vector ...
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Taylor expansion on a Manifold

Is there a way to define the Taylor expansion of a function $f:\mathcal{M}\rightarrow\mathbb{R}$, where $\mathcal{M}$ is a smooth manifold? I'm looking for a free coordinate definition. I guess it is ...
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1answer
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What is $dudv$ in the metric tensor?

In the definition of a $m$-dimensional Riemannian manifold $(M,G)$, if $(U;u^i)$ is a local coordinate system of $M$, the tensor field $G$ on $U$ can be written as $$ G = g_{ij}du^i\otimes du^j\;\tag{...
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Why do I need holomorphically immersed submanifold $M\to P^m(C)$ instead of embedding to use notion of hyperplane section bundle?

"Moreover, a holomorphically immersed submanifold $f:M\to P^m(C)$(i.e. $m$ dimensional projective space over $C$) has an induced bundle $f^\star(H)$($H$ is hyperplane section bundle over $P^m$ or dual ...
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Smoothness of a homomorphism on Lie subgroup

Suppose $\phi:G_1\to G_2$ is a Lie group homomorphism where $G_1$ is connected. Let $\phi(G_1)\subseteq G_3$ and $G_3$ is a Lie subgroup of $G_2.$ I want to prove that $\phi:G_1\to G_3$ is a Lie group ...
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Geodesic equation without metricity

If we relax the metricity condition, what change will be seen in the geodesic equation ? For instance, when the torsionless connection is relaxed while keeping the metricity intact, we see that the ...
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Isomorphism of the space of mixed (k, l+1) - tensors

Let $V$ be a finite-dimensional vector space. There is a natural isomorphism between $T^{k}_{l+1}(V)$ and the space of multilinear maps $$V^{*l}\times V^{k} \rightarrow V$$ I have found an ...
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Integrate over $D^2$, polar coordinates and other problems

Consider in $\mathbb{R}^3$ the unit sphere $$S^2 = \{ (x_1, x_2, x_3) \in \mathbb{R}^3 | x_1^2 + x_2^2 + x_3^2 = 1 \}$$ with inclusion map $i: S^2 \hookrightarrow \mathbb{R}^3$. Let $\alpha := x_1dx_2 ...
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Derivatives of surface curvature

I have an intuitive understanding of Gaussian curvature Mean curvature Minimum principal curvature Maximum principal curvature and how these at any given point on the surface can be expressed as (...
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1answer
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A compact hyperbolic manifold and fundmental group [closed]

I just start to learn differential geometry and have a problem. Let $M$ be a compact hyperbolic manifold, how to prove Z $\oplus$ Z is not a subgroup of $\pi_1(M)$ .
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Torus bundle over sphere

Let $G$ be a Lie group and let $T\cong\mathbb C/\mathbb Z^2$ be a compact complex torus. What is an example of a nontrivial $G$-fiber bundle $$T\hookrightarrow E\to S^2\; ?$$
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Is the parameterization of a regular surface a local diffeomorphism?

Let $\left\{ U_{\alpha },\varphi _{\alpha }\right\}$ be a local chart of a regular surface S $\subset \mathbb{R}^{3}$. That is: $\varphi_{\alpha}:U_{\alpha}\subseteq\mathbb{R}^2\rightarrow \varphi_{\...
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1answer
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Confusion about the Definition of Smooth Functions on a Manifold

I am slightly confused about the definition of smooth functions on a smooth manifold given in An Introduction to Manifolds by Loring Tu (Second Edition, page no. 59). The definition is given below. I ...
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Geodesically complete translation-invariant torsion-free affine connections on $\mathbb{R}^n$

Let $\nabla$ be a torsionfree affine connection on $\mathbb{R}^n$ which is invariant under the group of translations in $\mathbb{R}^n$. What are necessary and sufficient conditions on $\nabla$ for it ...
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1answer
31 views

Is there a way to write 1-forms to matrices?

Consider the following Riemannian metric on $\mathbb{R}^2$: $$g=(1+y^2)dx\otimes dx+xy(dx\otimes dy+dy\otimes dx)+(1+x^2)dy\otimes dy $$ If $w^1$ and $w^2$ are 1-forms defined by $$w^1=\sqrt{1+y^2}dx+...
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Haar integral invariance against inversion

Suppose $G$ is a compact Lie group, and $\omega$ is a left-invariant volume form over $G$ such that. $$\int_{G}\omega = 1$$ with respect to the orientation determined by $\omega$. If I define $$\...
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1answer
39 views

Interesting examples of submersions that are not surjective

What are some "interesting" examples of submersions that are not surjective? Usually, the notion of submersion comes with a prefix of surjective. Most of the maps we come across when we do ...
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Differential Equation that do not contain “y” directly.

I've been on this question for two days and I can't seem to get past the last integral. The natural logarithm of y seems to be a problem $\mathrm{ y\frac{d^2y}{dx^2} + \frac{dy}{dx} + \Bigl(\frac{dy}{...
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I need to evaluate $\iint_S xdydz + (x+y)dz dx+(x^2+2z)dxdy$

My problem asks me to evaluate the integral (using direct integration) $$\iint_S (x)dy\wedge dz + (x+y)dz\wedge dx+(x^2+2z)dx\wedge dy$$ Being $S$ the surface of the solid limited by the following ...
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2answers
73 views

$f=g \implies \nabla f = \nabla g$?

I want to disprove the following statement: $$f=g \text{ on } S \implies \nabla f = \nabla g \text{ on } S$$ where $S$ is some smooth closed surface and $f,g$ are smooth. I don't understand why this ...
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36 views

Is there any minimal surface that is topologically a sphere?

I am currently studying about Willmore energy and out of my expectation, I produced a simple result that any surface that is topologically a sphere (genus 0) cannot be a minimal surface. (I am not ...
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Is the osculating circle at a point of maximum curvature enclosed by a convex closed curve?

I'm trying to solve the following problem: (1) Given a convex, closed plane curve $\alpha$ without cusps that encloses an area $A$, prove that there exists a point $t$ such that the (unsigned) ...
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25 views

In a foliation due to a submersion, can we say the complements of leaves are also submanifolds?

I am very new to topology and geometry, so please excuse me if my statements are crude. According to https://en.wikipedia.org/wiki/Foliation#Submersions, if $f: \mathbb{R}^n \to \mathbb{R}^p$ is a ...
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How many degrees of freedom does a symplectic form have?

A symplectic 2-form on a $2n$ dimensional manifold has to be closed, nondegenerate and antisymmetric. My question is: Do these conditions imply how many degrees of freedom the symplectic form has? I ...
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15 views

Trouble with structure sheaf, derivations, tangent space

I'm teaching myself differential geometry, and the author I've been reading opts to work with germs of locally defined functions on a (not necessarily smooth) manifold to describe the tangent space. ...
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1answer
46 views

$S^2$ as a totally real submanifold of $\mathbb{CP}^1\times \mathbb{CP}^1$

Can the sphere $S^2$ be embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a totally real submanifold?
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1answer
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Curvature and Jacobian

We may describe the curvature of an object at a point by the Jacobian determinant of its Gauss map at that point. However, it seems to me that this might depend on the parametrization of the figure ...
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Definition of Poisson Bracket

Context: Let $f,g :T^*M\rightarrow \mathbb{R}$, the Poisson Bracket was defined classically as $$\{f,g\}=\sum\limits_{i=1}^n\frac{\partial f}{\partial q^i}\frac{\partial g}{\partial p_i}-\frac{\...
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0answers
28 views

A Problem from Docarmo Differential Geometry chapter 2.4

It's number 23 in chapter 2-4(The tangent plane). The Problem is, Let $P: C \rightarrow C$ be the complex polynomial $P(\phi)=a_0\phi^n +a_1\phi^{n-1}+ \cdots +a_n $, $a_0 \not= 0$, $a_i \in {C}$...
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1answer
25 views

Computation of the second derivative of the Jacobian of the change of the coordinates

I am trying to understand how to derive the equality $1.135$ of the book "A course in minimal surfaces" by Colding and Minicozzi. I derive $$(g^{ij}g_{ij}')' = (g^{ij})' g_{ij}' + g^{ij} g_{ij}''$$ ...
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Geometry of the complex Gauge group

This is a pretty naive question: Let $E\rightarrow X$ be holomorphic vector bundle on a complex manifold $X$. Denote by $\mathcal{G}=\Gamma(Aut(E))$ the group of complex smooth automorphisms of $E$. ...
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1answer
19 views

Immersion of tangent bundle in coordinates

Consider an open set $W\subset \mathbb R^d$ and a $C^\infty$ class function $f:W\to \mathbb R^{d-c}$. If $r$ is a regular value, $Z:=f^{-1} (r)$ is a $c$-dimensional submanifold of $W$, since $$\...
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1answer
24 views

Average of a tensor with respect to a group action

Consider a smooth manifold $M$ (assume it boundary-free and orientable) and a tensor field $\mathcal{G}\in\Gamma(\otimes^hTM\otimes^kT^*M)$. Let $\Phi:\mathbb{T}^p\times M \rightarrow M$ be a torus ...
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1answer
41 views

Confusion about the gradient term in the directional derivatives of a vector

Definition: Let $f$ be a differentiable real-valued function on $\mathbb{R}^3$, and let $\mathbf{v}_p$ be a tangent vector to it. Then the following number is the derivative of a function w.r.t. the ...
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1answer
21 views

A question about regularity of parameterizations of a surface [closed]

Can someone give an example of two permetrazations (1-1) of a surface that satisfy: At a same point (on the surface) one permetrazation is regular but the other is not regular
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First and second fundamental form of a surface given a Monge's parametrization

Let be a function $f:(u^{1},u^{2})\rightarrow f(u^{1},u^{2})\in\mathbb{R}$ of class $r\geq 2$ defined in a open $U\subset\mathbb{R}^{2}$. Let be a Monge's parametrization: $X:u\in U\rightarrow (u,f(u)...
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1answer
51 views

Can computers aid in solving problems of Topology/Differential Geometry

I was wondering whether computers can, or have been of any aid in fields such as topology and differential geometry. Usually when I think of computers, only problems of "finite"/"discrete" nature come ...
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29 views

Could covariant tensor fields which are **not** alternating be integrated across a manifold?

In differential geometry, we study the integration of differential forms, which are alternating covariant tensor fields. "Alternating" means that swapping two inputs leads to a sign change. For ...
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29 views

If the shape of a box is a cardioid, and we have the area of the cardioid and the height of the box, how do we find the volume? [closed]

So we are given the area of the cardioid and it is 75/2(pi) and we are given the height of the box which is 1 inch. How are we supposed to find the volume of the box? The cardioid that the box's shape ...

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