# 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.

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### Integration by parts with $|\nabla|:=\sqrt{-\Delta}$

What conditions on $f,g$ do I need to justify the integration by parts $$\int f|\nabla|g\,dx=\int(|\nabla|f)g\,dx.$$From $|\nabla|:=\sqrt{-\Delta}$ we have formally that $|\nabla|$ is a self-adjoint ...
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### The equivalence relation when constructing the associated bundle

When constructing from a typical fibre $F$, an action of a Lie group $G$ on that fibre and a $G$-principal bundle $P$ over some base space $M$ the associated bundle $P[F]$, one does this by ...
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### A centre of a simple closed planar curve

Let $c: [0,1] \rightarrow \mathbb{R}^2$ be a continuous planar curve such that $c(0)=c(1)$. I was wondering if there is a way to find something like a "centre of mass" of planar curve $c$. ...
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### Image of a $C^1$ manifold

Let $F: \mathbb{R}^n \to \mathbb{R}^{n+k}$ be a $C^1$ map with $n, l >0$. It's well know that $F(\mathbb{R}^n)$ is not necessarily a $C^1$ manifold of $\mathbb{R}^{n+l}$. Then what about the ...
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### Given $F: \mathbb R^n \to\mathbb R^m,$ $n < m,$ if $F$ has no critical points in a subset $U,$ then is F necessarily injective in $U?$

Given a map $f:\mathbb R^n \to\mathbb R^m,$ $n<m,$ and a connected subset $U$ of $\mathbb R^n$ where $F$ has no critical points, is this a sufficient condition for $F$ to be injective in $U$? Would ...
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### How do I show that the region {r > 2m} of the Schwarzschild spacetime is globally hyperbolic?

How do I show that the region {r > 2m} of the Schwarzschild spacetime is globally hyperbolic? I am just starting with this so I don't really know how to lay out this arguments. I consulted Beem's ...
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### What does it mean if we call a vector field holonomic?

I would like to know a simple and intuitive way to understand "the vector bundle is holonomic". If it is known that two vector fields X and Y are holonomic, then is it true that their direct ...
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### Projection from product of submanifolds is smooth

Consider the following: Let $X_1 \subseteq \mathbb{R}^n$, and $X_2 \subseteq \mathbb{R}^m$ be submanifolds of dimension $k$ and $l$. I have already shown that $X_1 \times X_2$ is also a submanifold. ...
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### Submanifold of matrix space

One can identify the space $M_{\mathbb{R}}(n,n)$ of real $n \times n$ matrices as $\mathbb{R}^{n^2}$. Consider the subset $S:=\{ A \in M_{\mathbb{R}}(n,n) : det(A)=1 \}$ and show it is a smooth ...
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### How to compute the first neumann eigenvalue of a spherical lune on the sphere?

Question: What is the first Neumann eigenvalue of Lune $L_{\beta}$ where $\beta$ denotes the opening angle of the Lune on the Sphere $S^2?$ Problem Setup: Parametrize the Lune in geodesic polar ...
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