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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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Tangent Bundle on S^3

how to show T(S^3) isomorphic to S^3 cross R^3? so can I say it for every odd dimension?I have shown it for n=1
user6493
6
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2 answers

Prove that a curve is spherical iff it satisfies the relation

I couldn't prove that a regular curve, such that the torsion and curvature never equal zero, satisfies the relation $$\frac{\tau}{\kappa}+(\frac{1}{\tau}(\frac{1}{\kappa})´)´=0$$ iff it's spherical, i.e, it lies entirely on a sphere.
user40276
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extension/"globalization" of inverse function theorem

I am curious as to what changes do we need to make to the hypotheses of the inverse function theorem in order to be able to find the global differentiable inverse to a differentiable function. We obviously need $f$ to be a bijection, and $f'$ to be…
user6421
6
votes
1 answer

What is the definition of a complex manifold with boundary?

Can anybody help me to be clear about this definition. I know the definition of a real manifold with boundary (as in Lee's book) and the definition of a complex manifold (locally diffeomophic to an open set in $\mathbb{C}^{n}$ and transition maps…
Binjiu
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How did Do Carmo get the following differential of the Gauss map?

Below is an example from Do Carmo's Differential Geometry page 139 "The Geometry of the Gauss Map". Let us analyse the point $p=(0,0,0)$ of the hyperbolic paraboloid $z=y^2-x^2$. For this, we consider a parametrisation $\textbf{x}(u,v)$ given by…
user338393
  • 1,165
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Abstract manifolds or do we need an ambient space?

I'm currently studying on J.Lee's "Introduction to smooth manifolds", but several other sources I consulted present the same line of thought. The most natural description of the $n$-dimensional sphere $S^n$ follows from imagining the sphere sitting…
Andrea Orta
  • 1,122
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Group action on manifold

I met a question as follows: Suppose $G$ is a finite group acting freely on a manifold $M$. Show the following: 1) $M/G$ is a manifold. 2) $H^i_{dR}(M/G)=H^i_{dR}(M)^G$. 3) if $N$ is a compact connected $n$- dimensional manifold which is…
Honghao
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Exponential of Lie Groups.

When the exponential map defined a bijection between the group G and their Lie algebra? The only example I know is the Heisenberg group.
Waldemar Barrera
  • 111
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How to prove that the wedge product is the determinant (Spivak's Claim)

In the book I am using now, Spivak's A comprehensive introduction to differential geometry volume 1 I have a question on page 205. Because he says the following: \begin{align*} \phi_1\wedge\cdots\wedge\phi_n &=…
user162343
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Is a straight line the shortest distance between two points?

Quite simply, I heard a lot of talk about how a straight line isn't necessarily the shortest distance between two points. Is this true, and if it isn't, how would that work?
Joseph
  • 275
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Question about the definition of the dual map of the differential

Background Let $M,N$ be smooth manifolds, $\psi: M \rightarrow N$ a $C^{\infty}$ and $M_m, N_{\psi(m)}$ the tangent spaces at $m \in M$ and $\psi(m) \in N$. The differential of $\psi$ at $m$ is the linear map $$d\psi: M_m\rightarrow…
recmath
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Something wrong with this do carmo exercise (1.3.3)?

I think I have found an unreported errata to this problem. Here is my attempt to solve the problem. Differential Geometry of Curves and Surfaces - Chapter 1 Section 3 Exercise 3 And here is the errata I can found online. In particular, I believe the…
Andrew Au
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The knowledge of $n=n(s)$ can be used to determine the curvature $k(s)$ and the torsion $\tau (s)$

Question: Show that the knowledge of the vector function $n=n(s)$ of a curve $\alpha:I\rightarrow \mathbb{R^3}$ with nonzero torsion everywhere, determines the curvature $k(s)$ and the torsion $\tau (s)$ of $\alpha$. Notes: $n$ is the normal…
shevaar
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Is integral curve a embedded 1 dimensional submanifold of the given manifold?

I can easily see a proof that shows its going to be an immersed submanifold . (I am removing the case if the vector field at that point is 0). I am not able to see if it's a embedded submanifold or not? Thank you.
Illuminata
  • 323
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How do I compute Gaussian curvature in cylindrical coordinates?

I just asked this question on ask.metafilter, and it was suggested that I ask here. Though I'm talking about coding something up, this question is about the math behind it, not the implementation. We have done analysis in the past where we've…
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