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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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Family of surfaces given their curvature

Let's suppose we know the shape operator (aka Weingarten operator) of a given surface everywhere in its domain. Is there any way, analytical or numerical, to find the family of surfaces having the given shape operator? And what if we know both first…
Pablo Rodríguez
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The measure of the set of lines that meet a convex closed curve is equal to its length

I wish to prove the following assertion: The measure of the set of lines that meet a convex closed curve $C$ (without multiplicites) is equal to the length of $C$. I know this is an application of Cauchy-Crofton theomrem, saying that: Let $C$…
Scorpio19891119
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Differential Geometry in R^3. Show there exists a unique unit speed circle given a unit speed curve.

Here is(are) the question(s). Let $f(t): (-t_0-\epsilon, t_0 +\epsilon)\to \Bbb R^3$ be a unit speed parametrized curve in $\Bbb R^3$. Suppose that $k(f(t_0)) > 0$, where $k$ is the curvature function. EDIT: I have a feeling the domain should be…
trystero
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Construction of an osculating circle

Let $\alpha$ be a unit-speed curve.Then there exists a unique circle $\beta$ such that $\beta(0)=\alpha(0), \ \beta'(0)=\alpha'(0), \ \beta''(0)=\alpha''(0).$ Attempt: Consider $\beta(s)= \textbf{p} +R\ \text{cos}(\frac{s}{R})\textbf{v}_1+R\…
Alexy Vincenzo
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Signed curvature and turning angle

I am trying to find the signed curvature of a function, I have so far that $$g'(t)=(\cos(\cosh(t)), \sin(\cosh(t)))$$ I know that $g'$ is unit speed so i don't have to parametrize by arc length, and I know that the direction of $g'$ is measured by…
J.c
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is the image of a polynomial map contractible?

There are often questions in differential geometry asking if a certain manifold (say a circle) has a polynomial parametrization. Are there topological obstructions to existence of such parametrization? More precisely, if we have a polynomial map…
adrido
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The tangency of two surfaces on a geodesic

If $S$ is a surface with a geodesic on it, can we find another surface $S'$ such that these surfaces are tangent on the geodesic with the additional condition that there is no other intersection? Furthermore, to what extent can we loosen the…
Summer
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Flow of a vector field: how existence of a flow line implies existence of flow.

I am unable to see why there exists $U$ such that $\phi_t(x)$ exists for all $t\in[0,T]$. Can you please help me to understand the argument above. Thanks.
Junu
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How can you express $f^*\left(\sum_{j_1,\dots,j_k} a_{j_1\dots j_k}dy^{j_1}\otimes\cdots\otimes dy^{j_k}\right)$ in terms of $dx^i$?

Suppose $f\colon M^n\to N^m$ is a map between manifolds, with $(x,U)$ and $(y,V)$ coordinates systems around $p$ and $f(p)$. How can you express $f^*\left(\sum_{j_1,\dots,j_k} a_{j_1\dots j_k}dy^{j_1}\otimes\cdots\otimes dy^{j_k}\right)$ in terms…
yunone
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Doubt on the definition of topological manifold

I need some clarifications on topological manifolds. My professor has defined them as second countable, connected, Hausdorff topological spaces which are locally homeomorphic to $\mathbb{R}^n_+$, where $\mathbb{R}^n_+=\{(x_1,\dots,x_n)\in…
batman
  • 2,065
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Proving that two sets are diffeomorphic

I have the following two sets $\mathcal{S}= \left\lbrace (x,y,z,w) \in \mathbb{R}^4 \mid x^2+y^2- z^2-w^2 = 1 \right\rbrace$ and $\mathcal{S}' = \left\lbrace (x,y,z,w) \in \mathbb{R}^4 \mid x^2+y^2- z^2-w^2 = r \right\rbrace$ for some non-zero…
Kerry H
  • 195
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Proof on metric tensor identity

What is the proof for this proposition: $$\frac{1}{2} g^{mi} \partial_k g_{mi} =\frac{1}{2g} \partial_k g $$
BinaryBurst
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Christoffel symbols proof

How do I proove this? I think it's something related to "index gymnastics"... $$\Gamma^i{}_{ki}=\frac{1}{2} g^{im}\partial_k g_{im}=\frac{1}{2g} \partial_k g =\partial_k \ln \sqrt{|g|} $$ Here's where I get…
BinaryBurst
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transition functions of a vector bundle

why are the transition functions in the definition of a vector bundle $P:E \rightarrow M$ termed as transition functions? they are $g_{\alpha \beta}:U_{\alpha} \cap U_\beta \rightarrow GL(n,\mathbb{R})$ and defined as $\phi_{\beta} \circ…
user103005
  • 361
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Can anyone explain how to calculate Gaussian and mean curvatures using B-Spline fit method?

I want to calculate Gaussian and mean curvatures of some real surface which is from some RGB-D camera like Kinect. So there is no a expression of this real surface. How to calculate the Gaussian and mean curvatures of it? A paper said using 3x3…
user116083
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