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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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Arbitrary dimensional object with constant ratio of volume to containing hyper cube's volume?

Let me introduce what I need by giving an example which is not working: n-sphere with radius 1 and respective hypercube with edges of length 2. The ratio reaches already in 10th dimension a level of less than 1%. My question essentially is whether a…
Raffael
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Is any foliation on a 2-torus induced by a suitable flow?

Consider the 2-dimensional torus $T^2=\mathbb{R}^2/\mathbb{Z}^2$, and a foliation on it (for example a foliation in circles, maybe the partition of the torus obtained form a Hopf-related map). I'm wondering if there are some condition on the…
fosco
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Does there exist a surface homemomorphic to a torus with positive Gaussian curvature?

This is a problem from the my last exam in Differential Geometry II and I didn't solve it. I'm studying again, but without success. So I need help. Does there exist a surface $S \subset \mathbb{R}^3$ which is homeomorphic to the torus…
Felipe
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Christoffel Symbols as Tensors

If you define a "generalized" Christoffel tensor as the following: $Chris(\omega; u, v) := (\nabla_u(\omega))(v) - (\tilde{\nabla}_u(\omega))(v)$ where $\omega$ is a dual vector, $u,v$ are vectors, and $\nabla$ and $\tilde{\nabla}$ are two covariant…
PPR
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Show suspension of differentiable n-manifold is (n+1) manifold.

I've been trying to solve a problem for a while. Let $W$ be an $n$-manifold and $F:M\to M$ be a diffeomorphism. The suspension of $F$ is defined by taking $M\times [0,1]$ and identifying every point $(x,0)\in M$ with $(F(x),1)$, and is denoted…
user124083
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Exterior product of n copies of 2-form

I have a problem with calculating exterior product of differential forms. Here is the problem: Let $\omega$ be a 2-form in $\mathbb{R}^{2n}$ given by $\omega=dx_{1}\wedge dx_{2}+dx_{3}\wedge dx_{4}+...+dx_{2n-1}\wedge dx_{2n}$. Calculate…
Alem
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Christoffel symbols equality

I have to prove the following: Let $\Omega \subseteq \mathbb{R}^d$ be open and $g$ a metric field on $\Omega$. For every $\phi \in \mathrm{Diff}(\Omega)$ let $\Xi^i_{jk}[\phi]$ be functions on $\Omega$ that transform in the same way as the…
michael
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The Definition of Convex Surface

Let $\Sigma$ be a $C^{\infty}$ compact surface in $R^3$. (1)If the tangent space of every point lies the same side of $\Sigma$, we call $\Sigma$ convex surface. (2)If the Guass Curvature $K>0$, we call $\Sigma$ ovaloid. (3)If $\Sigma$ is…
gaoxinge
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Length of a Continuous Curve

Let f be a bounded real valued function on [a,b], then define the length of the curve L(f) as follows: \begin{equation}\tag{1} L(f)=sup\sum\limits_{i=1}^{n}\sqrt{ (x_i-x_{i-1})^2+(f(x_i)-f(x_{i-1}))^2}, \end{equation} where sup is taken over all…
Sedergine
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Wald's definition of parallel transport

I was unsure whether to ask this here or at a physics SE. Wald's "General Relativity" defines parallel transport as follows: $\nabla$ is a derivative operator (is linear, obeys Leibniz rule, commutative with contraction, torsion free and is…
Diego
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Confusion with definition of a manifold.

I'm reading some notes about integration of differential forms and at the begining the author claims: A $1$-manifold in $n$ dimensions is just a curve parametrized as $X: (a, b) \subseteq \mathbb{R} \rightarrow \mathbb{R}^{n}$ (Me: plus other…
Antonio Velez
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The approximation of star-shaped domain

Let $\Omega$ be a star-shaped domain in $\mathbb R^n$, that is, $\Omega$ is an open set such that for any $x\in \Omega$, $tx \in \Omega $ for $0 \leq t\leq 1$. Can we find a sequence of star-shaped domains $\{\Omega_i\}_{i\geq 1}$ such that…
Summer
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Regular curve which tangent lines pass through a fixed point

How to prove that if a regular parametrized curve has the property that all its tangent lines passs through a fixed point then its trace is a segment of a straight line? Thanks
Richard
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planar curve if and only if torsion

Again I have a question, it's about a proof, if the torsion of a curve is zero, we have that $$ B(s) = v_0,$$ a constant vector (where $B$ is the binormal), the proof ends concluding that the curve $$ \alpha \left( t \right) $$ is such that…
Daniel
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What can we know about the metric if we know the Christoffel symbols (2 dimensions)?

Assume $U\subset {\mathbb ℝ}^2$, $g$ and $h$ two metrics on $U$. Assume that the Christoffel symbols $Γ^i_{jk}(g)\equiv Γ^i_{jk}(h)$, as a pointwise identity, for all sets of indices. Does it follow that $g=h$? If not, is there a counter-example?…
user105107
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