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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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Why does these curves are different at the origin.

Let $\alpha:\mathbb R\to \mathbb R^2$ be given by $\alpha(t)= (t^3,t^2)$. The trace of $\alpha$ is drawn below: Since $\alpha'(t)=(3t^2,2t)$, we have in $t=0$: $\alpha (0)=(0,0)$ and $\alpha'(0)=(0,0)$, then at the origin the tangent vector is…
user108205
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Showing that $S^1$ is orientable

I have a very silly question. While showing that $S^1$ is oriented we use two stereographic projection from the north and south pole. I have the atlas and everything. However, I just could not figure out how to obtain the Jacobian which idefine as…
Lilith
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Smooth map from $S^1$ to $S^1$ with degree zero

Is there any example of a smooth map $f:S^1\to S^1$ that has degree zero that is not the constant map? Either the map would have no regular values or every regular value would have an even number of pre-images with cancelling degrees, but I'm…
user114326
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norm of tangent to geodesic is constant

How do you prove that $g(T, T)$ is constant along a geodesic, where $g$ is a metric and $T$ is the tangent vector to the geodesic?
PPR
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What is the relationship between chain homotopy and the Leibniz rule?

Suppose that $f, g$ are chain maps between chain complexes, and $P$ an operator that takes $k$-chains to $k+1$-chains. Let $\partial$ be the usual boundary operator. Then the condition $$\partial P + P\partial = g - f$$ is that $P$ is a chain…
user801923
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Computing the area and length of a curve

Using the Riemannian hyperbolic metric $$g = \frac{4}{(1-(u^2+v^2))^2}\pmatrix{ 1 & 0 \\ 0 & 1 \\}$$ on the disk $D_p = \{(u,v)\ | \ u^2 +v^2 \le p^2\}$ compute the area of $D_p$ and the length of the curve $\partial D_p$.
Lays
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On a remark in Foundations of mechanics, 2nd Edition, by Abraham and Marsden

Given a $2$-form $\omega$ on a manifold $M$, let us denote by $N$ the kernel of $\omega$, i.e. $N:=\{u\in TM : \omega(u,\cdot)=0\}$. Their Proposition 5.1.2 shows that if $\omega$ has constant rank (and is closed) then $N$ is a tangent distribution…
agt
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local property of a curve to deduce a global property

I have a question about some facts about curves. Consider a curve such that all the normals intersect at a fixed point. Prove that the trace of the curve it's contained in a circle. And prove that if a regular curve has the property that all the…
Daniel
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How to see the Whitehead continum is a Cantor set?

In the construction of Whitehead manifold, a 3-manifold, open noncompact and contractible but not homeomorphic to $\mathbb R^3$, Whitehead used a set of nested tori. I can understand the construction in this way, but how to see the limit set is a…
Sun
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Space-filling sine

I'm trying to have a sine wave of freq. b follow another sine wave of freq. a. Where the wave I'm wrapping follows the other's frenet frame. I'm looking to repeat this process with the frequency doubling and the amplitude halving for each successive…
Anon42
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Holder estimate for eigenfunctions of Laplace operator on sphere?

I would like to ask if there is a holder esitimate for eigenfunctions of Laplace operator on sphere? I mean the esimate for \begin{equation} \|\partial^{\alpha}h_n\|_{L^{\infty}(S^{d-1})} \end{equation} in term of $n$, where $h_n$ is the $n^{th}$ …
user111209
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How to calculate $\mathrm{div}(f \hspace{2pt} \mathrm{grad} \hspace{2pt} g)$?

I'm a little confused about this: if $f, g: \mathbb{R}^{3} \rightarrow \mathbb{R}$ are differentiable functions, calculate $\mathrm{div}(f \cdot \mathrm{grad} \hspace{2pt} g)$. I try expanding definitons but I get stuck at $\mathrm{div} \hspace{2pt}…
Lucas Willy
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A question on the curvature of a regular parametrized curve

The next question is from Do-Carmo's baby book, page 30 question 3 in section 1-6. The question goes as follows: Show that the curvature $k(t)\neq 0$ of a regular parametrized curve $\alpha : I\rightarrow \mathbb{R}^3$ is the curvature at $t$ of of…
MathematicalPhysicist
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Decomposition of linear partial differential operators

I was wondering about the following: Let $M$ be a smooth, second-countable (possibly noncompact) manifold and let $E$ and $F$ be smooth vector bundles over $M$. Can every smooth linear partial differential operator $P$ from $E$ to $F$ be written…
Marcel de Reus
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Singular points on parallel surfaces

Let ${\bf r}(u,v)$ be a parameterization for the surface $M$ in $\mathbb{R}^3$. If we denote by ${\bf N}$ the unit normal vector field of $M$, we can define a parallel surface $M_{d}$ in the following way: \begin{equation} {\bf r}_{d}(u,v) = {\bf…
Ben Perez
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