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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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A question about smooth invariance of domain

Theorem 22.3 (Smooth invariance of domain). Let $U \subset\mathbb{R}^n$ be an open subset, $S \subset\mathbb{R}^n$ an arbitrary subset, and $f : U \rightarrow S$ a diffeomorphism. Then $S$ is open in $\mathbb{R}^n$. I can't understand why the set…
Toney Leung
  • 739
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Length of differential element along a cone.

If I calculate the differential length $ds$(see picture) as $rd\varphi$ I get a different value than if I take the projection of $r$ and the projected angle $d\theta$ to get $Rd\theta$ (where $R$ is the projection of $r$). Does $d\varphi$ not…
Akshat Sharma
  • 476
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A vector field whose Jacobian(?) is the identity is an Euler field?

I have a connection $\nabla$ in the tangent bundle of a manifold $M$, and a vector field $v$ on $M$ satisfying $\nabla_w(v)=w$, for all vector fields $w$ on $M$. Let $P\in M$ be a point where $v$ vanishes. My hope is that I can find an…
Mathi Shard
  • 121
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Let $S$ be a (n embedded) submanifold of a manifold $M$. Let $\Phi(M)$ be a diffemorphism. Is $\Phi(S)$ an embedded submanifold?

I feel like the answer to the above question is "yes", as one should just be able to compose the embedding with the diffeomorphism to get an embedding, but I would appreciate confirmation, as I still find some of this stuff a bit tricksy
Diarmuid Dempsey
  • 103
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Euler characteristic of a compact surface

Determine the Euler characteristic of the surface $$ M=\left\{(x,y,z); \sqrt{x^2+y^2}=1+z^{2n}, 0< z< 1\right\} $$
swallenberg
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Calculating gaussian curvature

I am stuck with this problem. The quadric surface $xy=z^2+z$ is given to me, and I have to find the gaussian curvature in $(0,0,0)$. Is there an easy way? I have tried parametrizing it but I was unable to do that. Can I go to the projective space,…
Extreme_Math
  • 786
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Volume of Ball on Spherical Manifold

Let $S^n$ denote the $n$-dimensional spherical manifold, represented by the sphere model. Let $B_x(r)$ be the ball centered at $x\in S^n$ of radius $r$. What I'd like to know is how the volume of $B_x(r)$ behaves asymptotically in terms of $r$. Is…
ndrizza
  • 1,348
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Question on the Gauss-Bonnet theorem

According to Wikipedia the global Gauss-Bonnet theorem concludes, $\int_{M}KdA + \int_{\partial M}k_{g}ds=2\pi\chi(M)$. The lecture notes I am using however does not have the integral with the geodesic curvature but they seem to have the same…
Number4
  • 397
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finding gaussian curvature of a metric $ds^2=du^2+\lambda^2 dv^2$

For a surface with the metric $ds^2=du^2+\lambda^2 dv^2$, where $\lambda$ is a function of $u,v$; prove that the Gaussian curvature is given by $\displaystyle{K=-\frac{1}{\lambda}\frac{\partial^2 \lambda}{\partial…
am_11235...
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Isometry between plane and cylinder

I am looking for an explicit isometry between the plane and a cylinder, is this possible to find?
Number4
  • 397
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A submanifold is embedded iff it satisfies the local k-slice condition

I'm having trouble understanding the proof in Lee of the above claim. More precisely, that if $M$ is a smooth manifold and $S \subset M$ is an embedded $k$-submanifold, then $S$ satisfies the local $k$-slice condition, i.e. for all $p \in S$, there…
staedtlerr
  • 486
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Norm of differential forms

I am trying to understand a detail in the following paper: Bott-Chern currents and complex immersions J.-M. Bismut, H. Gillet, and C. Soulé https://projecteuclid.org/euclid.dmj/1077297147 On page 262, they write: Let $C^1(M)$ be the set of…
hwong557
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Creating a new local frame using a map $F:U \rightarrow GL(k, \mathbb{R})$

In the proof for the following theorem: where we defined a vector bundle to be flat if it is locally trivial to $M \times \mathbb{R}^n$, so there exists a parallel local frame. The rest of the proof is clear to me, but I don't understand why…
eager2learn
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Hypotheses for existence of Tubular neighborhoods

Let $M$ be a submanifold of $N$, i.e. there is a smooth map $i:M\rightarrow N$ which is a topological embedding and whose differential is everywhere invective. I call tubular neighborhood of M in N an open neighborhood of $i(M)$ that has the…
Angelo Brillante Romeo
  • 569
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How does the Evolute of an Involute of a curve $\Gamma$ is $\Gamma$ itself?

How does the Evolute of an Involute of a curve $\Gamma$ is $\Gamma$ itself? Definition from wiki:-The evolute of a curve is the locus of all its centres of curvature. That is to say that when the centre of curvature of each point on a curve is…
user464147
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