Questions tagged [differential-geometry]
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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.
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Coordinate basis
A tensor article I read describes the old basis as xi and the new basis as ya and the coordinate transformation as ya(xa). It then says the functions are invertible so you can obtain the function xi(... linear-algebra differential-geometry tensors- 1
deg of $f$ is well defined
I was reading Guillemin & Pollack differential topology book, there is a corollary in page 115 that I can't work it out: Corollary. If $\operatorname{dim} X=\operatorname{dim} Y$ and $Y$ is ... differential-geometry manifolds differential-topology- 3,017
Area of Complete Hyperbolic surface
I'm studying the area of hyperbolic surfaces and have reached a proposition that is not understandable.I really appreciate it if you could help me with it. Proposition: A complete hyperbolic surface F ... differential-geometry algebraic-topology hyperbolic-geometry- 1
Two differential forms are the same if they are the same locally.
Two differential forms are the same if they are the same locally. Why is it the case? Could anyone give me some suggestions in this regard? Thanks for your time. differential-geometry differential-topology differential-forms- 143
Non-trivial $P^1$ bundle over complex tori
I am studying holomorphic $\mathbb P^1$-bundle over complex tori $\mathbb C^2/\Lambda$. There should be many non-trivial examples, but I can't write one explicitly. So I wonder if there are any ... differential-geometry algebraic-geometry complex-geometry fiber-bundles- 57
Prove that $f^*(a)$ is a regular surface.
Let $f : U ⊆ \Bbb R^3 → R$ a differentiable function on $U$ an open of $\Bbb R^3$. If $a ∈ f_∗(U)$is a regular value of $f$, so $f^∗(a)$ is a regular surface in $\Bbb R^3$. Try: Let $p=(x_0,y_0,z_0)\... differential-geometry surfaces- 701
Show that $M :=\{J\in M_{2n}(\mathbb{R}); J^2 = -I\}$ is a submanifold of $M_{2n}(\mathbb{R})$
I want to show that: $$ M := \{J \in M_{2n}(\mathbb{R}) \mid J^2 = -I \} $$ is the submanifold of $M_{2n}(\mathbb{R})$, and I am given a hint to use Theorem 1. Theorem 1. Let $m,n,l \geq 0$. Suppose ... differential-geometry smooth-manifolds geometric-topology- 31
Torsion Free Spin Connection
Ok I am not exactly sure how much of this common notation/terminology, and how much is unique to the book I'm reading, so bear with me for a moment here. First we have a vector bundle $E$ associated ... differential-geometry differential-topology connections gauge-theory cartan-geometry- 826
Second Fundamental Form of the Graph of a Function of Higher Codimension
Let $f:\mathbb{R}^n\to\mathbb{R}^m$, $m\geq 2$, be a smooth function. I would like to find an explicit description for the second fundamental form of the graph of $f$ in terms of the Hessians and ... differential-geometry submanifold- 121
How to show that if two vector fields $X$ and $Y$ are tangent to a submanifold then so is their Lie bracket?
Let $M$ be a smooth manifold and let $Q$ be a submanifold of $M$. Consider two vector fields $X, Y \in \mathfrak{X}(M)$ such that $X_q, Y_q\in T_qQ$ for every $q\in Q$. Prove that $[X, Y]_q\in T_qQ$ ... differential-geometry vector-fields lie-derivative- 2,542
Show that $X=( x_1,x_2,x_3,x_4,x_5)\in \mathbb{R^5}:x_1^4+x_2^4=1+x_3^2+x_2^2+x_5^2$ is a regular orientable manifold of dimension 4
The question asks to prove that the orinetability of the manifold. Using regular level set theorem we can prove that it is 4 dimensional manifold but how to prove it's orientable or not? For that do ... differential-geometry differential-topology submanifold- 21
Injectivity, surjectivity, and localization of $C^\infty$-rings
In classical commutative algebra, a ring morphism $f:A\to B$ is injective/surjective iff it is so Zarsiki locally on its source. That is, iff there are elements $a_i\in A$ so that $1\in(a_i)$ and all $... differential-geometry commutative-algebra synthetic-differential-geometry- 300
Measure of null (or full) subsets of manifolds and absolute continuity of smooth measures
It is common to define null (resp. full) subsets of a $n$-dimensional manifold $M$ in the following way: $A \subseteq M$ is a null subset if its preimage under every coordinate chart is a null (resp. ... measure-theory differential-geometry- 107
Structure induced by a sheaf
Let $n\geq 1$ and $F$ be a sheaf on $\mathbb{R}^n$ of continuous functions. Assume for every open subsets $U,V$ of $\mathbb{R}^n$ and every $f\in F(U)$, $g\in F(V)$, $g\circ f\in F(U\cap f^{-1}(V))$. $... differential-geometry manifolds sheaf-theory riemann-surfaces- 41
Why is $f_t= \gamma^{'}_s$?
I am currently working on the first variation formula of energy and I use this file here to understand this notion.. We are working on the following construction: Let $\gamma : [a,b] \to M$ be a ... differential-geometry riemannian-geometry calculus-of-variations- 797
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