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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.

Differential geometry is a branch of mathematics that uses techniques in calculus and linear algebra to study geometry. It is closely related to differential topology and PDEs (geometric analysis).

32835 questions
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The gradient as a special case of the differential (or push-forward)

I am tripping over something elementary (I think). Given a smooth map $f\colon M\to N$ between smooth manifolds, the differential of $f$ at $p\in M$ is defined as \begin{align}\mathrm{d}_pf \colon T_pM &\to T_{f(p)}N\\ X&\mapsto X(-\circ…
prt13463
  • 355
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Connection matrix for a vector bundle

Let $E \rightarrow M$ be a vector bundle. In the book Differential Analysis on Complex Manifolds by R.O. Wells a connection on E is defined to be a map a linear map $D: \Omega(E) \rightarrow \Omega^1(E)$ such that $D(\phi s) = d\phi\otimes s + \phi…
Qu4nt4
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Any fiber bundle is in fact principal fiber bundle.

I wonder whether or not ${\bf R}$ is a topological group. In fact ${\bf R}$ is a group with addition structure. So is it topological group ? In fact any ${\bf R}^n$ is a topological group. Note that any element in ${\bf R}^n$ is mapped into…
HK Lee
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Peeling the hyperapple

The volume of a $d-$ball of radius $R$ is $$V_d(R) = C_d R^d$$ with $C_d$ a constant depending on $d$. This means that $$\frac{V_d(R-\delta)}{V_d(R)}= \left(1 - \frac{\delta }{R} \right)^d$$ where we choose $\delta \in (0,R)$. For every choice of…
valerio
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Constructing a smooth bump function on a manifold

In "Loring W. Tu, An introduction to manifolds" the following question exists: Let $q$ be a point of an $n$-dimensional manifold $M$ and $U$ any neighborhood of $q$. Construct a smooth bump function at $q$ supported in $U$. I answered that question…
Hussein Eid
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Which planar curve has curvature linearly on arc length?

Which planar curve has natural equation $ k(s) = a*s $ ? where k(s) is curvature function on arc lenght parameterization and $ a \neq 0$
halfpog
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Another formulation of existence theorem for smooth bump functions

Let $M$ be a smooth manifold (at least Hausdorff). Most of the times, as an application of the existence of partitions of unity on $M$, the existence of smooth bump functions is shown (see for example Lee or Warner). It is stated like this: For any…
TheGeekGreek
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Connection on the plane: examples?

I am having trouble understanding how to compute the connection on a manifold. I see the definition $\nabla_ie_j=\Gamma^k_{ij}e_k$, but I am not sure if this equation is supposed to define $\nabla_i$ or $\Gamma$. If it defines $\Gamma$, then how is…
thedude
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Holonomic basis and commutativity

I am trying to better understand the concept of connection in diff. geometry by defining different vectors fields and working with them. For example, if I take the usual coordinates $x$ and $y$ coordinates on the plane, I have the vector fields…
thedude
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$C^2$ isometric embedding of the flat torus into $\mathbb{R}^3$

What is the reason that there is no $C^2$ isometric embedding of the flat torus inside $\mathbb{R}^3$? Is there an explicit proof of this fact anywhere? The flat metric must violate some condition for the $C^2$ isometric embedding. And as we know,…
Ayan
  • 315
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Does diffeology provide moduli for classical constructions?

Do classical constructions on differentiable manifolds like affine connections, Riemannian metrics, or (almost) complex structures have moduli spaces in category of diffeological spaces?
Aleksei Averchenko
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Difference between sphere and geodesic sphere and hypersphere

I see the Alexandrov theorem wich say that every comapct, without boundary embedded in Euclidien space must be a round sphere. Can someone explain for me the difference between geodesic sphere rond sphere and hypersphere? Thank you.
Mohammed Mohammed
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Vector field on torus as a submanifold of $\mathbb R^4$

Let $f(\theta,\phi)=\frac{1}{\sqrt{2}}(\cos \theta,\sin \theta,\cos \phi,\sin \phi)$ be immersion of torus into $\mathbb R^4$. How to prove that $\nabla_{\frac{\partial}{\partial \theta}} \frac{\partial}{\partial \theta}=0$? It should be something…
dmm
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Transition functions on a quotient manifold

Here's an exercise given during a course in Differential Geometry that I'm taking. Let $M$ denote a smooth manifold and let $G$ be a finite group of diffeomorphisms acting on it without fixed points (that is, $g(p)=p$ for some $p\in M$ forces $g$…
Giuseppe Negro
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Difference of vectors living in different tangent spaces

I have a question about tangent vectors of manifolds. Imagine that I have a vector $V$ living in $T_pM$ and $W$ in $T_qM$. In my book it is written that the difference between those vectors is ill defined. I would like to really understand…
StarBucK
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