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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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Intuition behind differentials on smooth manifold

Let $f: M \rightarrow \mathbb{R}$ be a smooth function on a manifold $M$. Then $df:M\rightarrow TM^*$ denotes a cut on the cotangent bundle. Then $df$ with $p \in M$ is given by $$ df(p) = \sum_{j=1}^{n}df(p)\left(\frac{\partial}{\partial x_j}\bigg…
user582360
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Isometry between surfaces

Let $S$ be the surface $S = \{(x,y,z) : x^2 + y^2/4 = 1\}. $ Minding theorem says that $S $ and $C $ the cylinder $x^2 + y^2 = 1$ are locally isometric, but can anyone build an local isometry between $S $ and $C $?
hal97
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A different structure in S1?

I want to know another structure in S1. I want that it not be diffeomorphic to the usual structure. Thanks!
hal97
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Lines in a regular surface passing though a point

The question is to prove that a regular surface $S \subset \mathbb{R}^3$ can't contain more than two lines passing though a point $p$ if its gaussian curvature at $p$ is $\neq 0$
José
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Why is that quantity a constant?

Help needed! What have I done wrong here? Given the metric $$ds^2 = dr^2+r^2d\theta^2$$ And $$R_{ij}=\nabla_i \nabla_j f(r)$$ where $\nabla_i$ is a covariant derivative, and $f=f(r)$ is a scalar function. I wish to show that $f'(r)\over r$ is a…
Harrold
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Harmonic extension of orientation preserving maps.

Let $u:\mathbb{S}^{n-1}\mapsto\mathbb{R}^n$ be an orientation preserving $(C^1)$ map. Is it true that it's harmonic extension $u_h:B_1\mapsto\mathbb{R}^n$, namely the map that is such that $\Delta u_h=0$ in $B_1$ and $u_h=u$ on…
KZ7
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Curvature of Regular curve at a point

So I was looking through Differential Geometry of Curves and Surfaces by Banchoff and Lovett and found a problem that I had some trouble with. I'm not quite sure what I'm missing, I haven't taken Differential Geometry in some time so I'm a bit…
user62931
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First fundamental form of a complex circle

I am trying to see how to compute the First Fundamental Form of this set: $\{z_1^2+z_2^2=1,(z_1,z_2) \in \mathbb C^2 \}$ . First I have seen in this post is homeomorphic to the sphere without two points. But then the parametrization I have seen…
energy
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Calculate the Frenet apparatus $\kappa,\tau,T,N,B$ where $T,N,B$ are the tangent, normal and binormal vectors.

Calculate the Frenet apparatus $\kappa,\tau,T,N,B$ where $T,N,B$ are the tangent, normal and binormal vectors. The curve i'm trying to calculate the Frenet apparatus for is given by: $\gamma(t)=(e^tcos(t),e^tsin(t),e^t)$ In the example we did in…
user624065
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Gaussian Curvature 2

Let $\varphi(u,v)=(u,v,h(u,v))$ a parametrization of graph $\Gamma_h$ of $h:\mathbb{R}^2\to \mathbb{R}$ . Show that the Gaussian curvature can be expressed as $K(u,v)=\dfrac{\text{det}(Hess(h))}{(1+\|\nabla h\|^2)^2}$, where $Hess(h)$ is the Hessian…
Julio_fmat
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Given a smooth manifold $M$, how does local coordinates affect the basis of $T_p(M)$?

Let's say I have a smooth manifold $M$ of dimension $n$ and a smooth chart $(U, \phi)$ consisting of open set $U$ of $M$ along with a homeomorphism $\phi : U \to \widehat{U} =\phi[U] \subseteq \mathbb{R}^n$. Recall that the component functions…
Perturbative
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Geodesic curvature

Clairaut’s theorem. How to prove that geodesic curvature parallel Radius of the surface of revolution (i. e the value inversed to geodesic curvature) is equal to tangential line segment and meridian closed between the point of tangency and surface…
Gera Slanova
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exterior derivative of the flow

Let $E(x, y, z) = (x, y, z)$ be a vector field on $\mathbb{R}^3$. $α(x, y, z) = x dy ∧ dz − y dx ∧ dz + z dx ∧ dy$ is a 2-form. Find $\phi^{t*}_E\alpha$. The flow of X is $\phi^t = (x_0e^t,y_0e^t,z_0e^t)$. I need to compute $d\phi_t^{*}$ as…
Conjecture
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Curvature of projected conic sections

who can prove an easy and beautiful observation on a sheet of paper in a few lines? I have used a computer algebra system to verify (the possible input is contained in the "proof") the following. Let $C\subset\mathbb R^3$ be a cone (quadric) with…
Stephan
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Cohomology of $\mathbb{S}^n$

I have a corollary stating: if X is contractible then $H_0(X) = \mathbb{R}$ and $H_n(X) = \{0\}$ for $n>0$. But $\mathbb{S}^n$ is contractible and $H_n(\mathbb{S}^n) = \mathbb{R}$. I must be missing something. Thank you!
Conjecture
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