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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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Atlas of a regular surface

I have the following set of $R^3$: $$ S=\{(x,y,z) \in \mathbb{R}^3: \, x^2+y^2-z^3=1\} $$ It is immediate to see that $S$ is a regular surface. How may I find an atlas? When $z \neq 0$, as $z=\sqrt[3]{x^2+y^2-1}$, we have the parameterization…
TheWanderer
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Crossing problem - a puzzle

I ran across the following problem in Barret O'Neill's Elementary Differenetial Geometry: "Let C be a Curve in the xy plane that is symmetric about the x axis. Assume C crosses the x axis and always does so orthogonally. Explain why there can be…
Gene Naden
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Proof that the image of a surface, under a diffeomorphism, is also a surface.

The problem: "If $F:R^3\rightarrow R^3$ is a diffeomorphism and M is a surface in $R^3$, prove that the image F(M) is also a surface. (Hint: If x is a patch in M, then the composite function F(x) is regular, since $(x)^*=F^*x*$.) My attempt at a…
Gene Naden
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Schoen estimates for stable minimal surfaces

I'm refering to article of R.Schoen 'Estimates for stable minimal surfaces in three dimensional manifolds' (1983). In the first paragraph of the proof of theorem 2 the author wants to apply the methods of article 4 (i think theorem 1 of article 4)…
user55449
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The open disk and closed disk are a regular surfaces

this is the exercice 2-2 of Do Carmo's page 67 1- is the set $\{(x,y,z)\in \mathbb{R}^3 ; \;z=0 \text{ and } \;x^2+y^2\leq 1 \}$ a regular surface ? 2-is the set $\{(x,y,z)\in \mathbb{R}^3 ; \;z=0 \text{ and }\;x^2+y^2<1 \}$ a regular surface ? my…
Bernstein
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Trying to prove $\partial^2=0$ on $k$-cells

Consider a $k$-cell $ c : [0,1]^k \to U \subset \mathbb{R}^n , (t_1,...,t_k) \mapsto c (t_1,...,t_k) $. Then the boundary of $c$, $\partial c$ is defined as $$ \partial c := \sum_{i=1}^k (-1)^{i-1} \left( c_i^1 - c_i^0 \right) $$ where $$ c_i^1 :=…
user577413
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Commutator of two Laplace operators

Let $M$ be a smooth manifold and $g,\bar{g}$ two Riemannian metrics on $M$. Let $\Delta$ and $\bar{\Delta}$ be the Laplace operators associated to $g$ and $\bar{g}$, respectively. When acting on a vector field $X\in \Gamma(TM)$, what is the…
srp
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Differentials and second order derivative

I have a question from my tutorials and I don't know how to start... Let $U$ be open in $\mathbb{R}^{n}$ and let $f:U\rightarrow \mathbb{R}$ a $C^{2}$ function. Let $p$ be a point in $U$ where $df_{p}$ of $f$ at $p$ does not vanish. Show that there…
enoughsaid05
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How to compute the index of a given vector field on a triangular mesh

Suppose that I have a triangular mesh (discrete surface composed of triangles). Now, I have been given a vector field (one vector with each triangle, tangential, unit length, so can be represented by only one angle). I would like to compute the…
mengda
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Embedded and immersed submanifold

I try to solve the following problem:For each $a\in \mathbb{R}$, let $M_a$ be the subset of $\mathbb{R}^2$ defined by $$ M_a=\{(x,y):y^2=x(x-1)(x-a)\}.$$ For which values of $a$ is $M_a$ an embedded submanifold of $\mathbb{R}^2$? For which values…
JYM
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How can I show what the induced metric is?

I'm unclear on how to solve any of these problems, can someone please help me with what to do? a) Let S be the upper portion of a cone in $\mathbb{R}^3$ parametrized by $\phi(u^1,u^2) = (u^1\cos u^2, u^1\sin u^2)$; $u^1 >0$. Show that the induced…
user130306
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Does the orientation on a product of manifolds depend on the order of the product?

I'm learning differential geometry for the first time. I'm sure this question is answered in a number of books, but it doesn't seem to be addressed in any book I have. Let $M,N$ be orientable manifolds of dimension $m,n$. I'd like to view an…
user355183
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Metric connection

How to prove that any vector bundle with a fiber metric g admits a metric connection? It seems I should use partition of unity, but I have no idea how to proceed. Also it seems there are two definitions of connection on a vector bundle E, one is…
user8400
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Is this an example of a regular surface with planar point that is strictly locally convex?

Definition of locally convex: (Definition) (Local Convexity and Curvature). A surface $S\subset \mathbb{R}^3$ is locally convex at a point $p\in S$ if there exists a neighborhood $V \subset S$ of $p$ such that $V$ is contained in one of the closed…
Ecotistician
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Inversion of Hopf's Umlaufsatz

Hopf's Umlaufsatz (better known as?) says: Let $\gamma$ be a simple closed differentiable positively oriented curve in the plane. Then for its curvature $\kappa$ it holds: $$\int_{\gamma}\kappa\ \text{d}s = 2\pi$$ I wonder if (and cannot see…
Hans-Peter Stricker
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