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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).

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$\deg(f)\neq 0\Rightarrow f$ is surjective

How can I prove the following statement? Let $f:M\rightarrow N$ be a smooth map between closed, connected, oriented manifolds of the same dimension. If $\deg(f)\neq 0$, then $f$ is surjective. Here $\deg(f)$ denotes the Brower degree of $f$. Note:…
Daniel Robert-Nicoud
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Hopf fibration with 7-dim. spheres as fibers.

I've read that one can generalize the Hopf fibration to get a fibration with 7-dimensional sphere fibers $\mathbb{S}^7 \rightarrow \mathbb{S}^{15} \rightarrow \mathbb{S}^8$. What is the explicit formula for this?
Sak
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Question about covariant derivative on manifolds.

I am currently learning about the covariant derivative on smooth manifolds with the following definition: Definition. A covariant derivative of vector fields is an operation that associates to two vector fields $X$ and $Y$ a new vector field…
QED
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What is the meaning of "without referring back to the ambient space $R^3$ where the surface lies"?

I'm reading Do Carmo's Differential Geometry book, here: What is the meaning of "without referring back to the ambient space $R^3$ where the surface lies"? Does it mean that we can compute it directly without appealing to some inverse mapping? If…
Red Banana
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How to use chain rule in this equation?

Source: Prof. Meinrenken's notes on Differential Geometry, page No 72: Theorem $4.1$. let $(U,\varphi)$ be a coordinate chart around $p$.A linear map $v :C^{\infty}(M) \to \mathbb{R}$ is in $T_pM$ if and only if it has the form…
wasiu
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Norm of geodesic velocity vector

The definition of affine geodesic is clear: a curve with covariant derivative respective to Levi-Civita connection $\nabla$ of velocity vector $\dot{\gamma}$ respective to vector field $\dot{\gamma}$, for arbitrary instant $t \in [0, 1]$ equal to…
Bruno Peixoto
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Is this the correct understanding of how a geometric surface works?

I'm reading "Elementary Differential Geometry" by Barrett O'Neill. Most of the book is spent looking at surfaces in $\mathbb R^3$, but eventually he introduces the "abstract surface", which I understand to be a surface $M$ which doesn't…
ham burglar
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Does a differential-geometry-isometry map a (sub)vector space to a (sub)vector space?

Given an isometric embedding of the tangent bundle $TM$ of a Riemannian manifold with Sasaki metric into $ℝ^N$ with standard metric. My intuition tells me that the vector spaces $T_pM$ ($p ∈ M$) are mapped to affine sub spaces of $ℝ^N$ but cannot…
flukx
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Do Carmo Problem - Section 3.3 - 12

The problem statement is as follows - Consider the parametrised surface $$ x(u,v) = \bigg(\sin u\cos v , \sin u\sin v , \cos u + \log(\tan\frac{u}{2}) +\phi(v) \bigg) $$ where $\ \phi\ $ is a differentiable function. Prove that a. The curves $\ v…
symmetric
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Is the following $2$-form exact? What about the restriction of $\omega$ to another surface?

I am not very good at working with $2$-forms, though I'm hoping to learn. I found this problem and have trouble starting it. I also do not know what a restriction would look like. Is the $2$-form $\omega = z\, dx \wedge dy$ an exact form in…
phenomenalwoman4
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Hyperboloid of one sheet has only one closed geodesic

Let $S : x^2+y^2-z^2 = 1 $ be the hyperboloid of one sheet. We know that there exists a closed geodesic, namely the unit circle $\{(\cos\theta, \sin\theta,0) , \theta \in [0,2\pi]\}$, since it is the locus of fixed points of the reflection along the…
samu
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Kenmotsu - rotational surfaces

I was wondering if someone knows which is the parameterization that we made in these cases. I calculated till $$Z(s)=\left(\frac{1}{2iH}(1-e^{-2iHs})+C\right)e^{2iHs},$$ but can't figure out what parameterization was done after that and why we have…
user981656
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What is the point of the idea of "contact" in differential geometry?

I've been having introductory lectures on differential geometry and we came to the idea of "contact". There are two definitions: Let $\alpha: I \to \Bbb{R}^3$ and $\beta: \overline{I} \to \Bbb{R}^3$ be regular curves such that $\alpha(t_0)=\beta…
Red Banana
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Diagonalization of the Ricci curvature tensor in dimension 3

It seems to be well-known that in dimension 3, we can simultaneously diagonalize the metric and the Ricci tensor at any given point (see https://mathoverflow.net/questions/80452/diagonalizability-of-the-curvature-operator). I am unable to prove this…
61plus
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Number of line of curvature meets at one point

I am now consider the surface given by $$f=(x,y,x^3-3xy^2)$$ And I have been asked to prove that there are three line of curvature meet at origan point. I can compute that \begin{align*} f_x&=(1,0,3x^2-3y^2)\\ f_y&=(0,1,-6xy)\\ f_x\times…
Emiya
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