# Questions tagged [dg.differential-geometry]

Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

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### Mean sphere in hyperbolic 3-space

What would be the natural equivalent of the notion of a mean sphere at a point of a smooth surface when the surface no longer lives in the Euclidean 3-space but in the hyperbolic 3-space ?

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### Ricci flow on locally symmetric noncompact manifold

As it is mentioned by Deane Yang in Ricci flow preserves locally symmetry along the flow, we know the local symmetry is preserved under the Ricci flow on the compact manifold since we have the ...

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### About Pogorelov-Nadirashvili-Yuan's local isometric embedding counterexample

In Pogorelov's paper "An example of a two-dimensional Riemannian metric admitting no local realization in E3. Dokl. Akad. Nauk SSSR Tom 198(1), 42–43 (1971); English translation in Soviet Math. Dokl. ...

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### Extension of Vector Field in the $\mathcal{C}^r$ topology

This question was previously posted on MSE.
Let $M\subset \mathbb{R}^n$ be a compact smooth manifold embedded in $\mathbb{R}^n$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is ...

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### Gram-Schmidt map as a Riemannian submersion

We equip $\mathrm{GL}(n,\mathbb{R})$ and $\mathrm{O}(n)$ with their left-invariant metrics, whose restrictions to the corresponding neutral elements is the standard inner product $\mathrm{Trace}(AB^{\...

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### Flattening a connection on a Kähler manifold

Say $M$ is a closed K?hler manifold and $(V, \nabla)$ is a (say) constant Hermitian bundle on $V$ with (say) trivial flat connection. Now $M$ K?hler gives several distinguished classes of closed one-...

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### How to compute the eta invariant of torus

I was wondering how to compute the eta invariant $\eta(T^3)$ of a flat torus $T^3$, with respect to the signature operator.
In general, how can we compute the $\eta(T^3/\Gamma)$ of a finite quotient ...

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### A homotopy problem for morphisms of dg-algebras

Let $\mathfrak g$ be a real finite-dimensional Lie algebra, and suppose we are given two morphisms of dg-algebras $f,g: (C^\bullet(\mathfrak g),d_{CE}) \to (\Omega^\bullet(\Delta^n),d)$, such that ...

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### Transnormal function and isoparametric function

Let M be a connected complete Riemannian manifold and denote by $\nabla$ and $\triangle$ the Levi-Civita connection and the Laplace operator of M, respectively. A non-constant function f of class $C^{...

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### The totally geodesic manifolds of 3-hyperbolic hypersurface [duplicate]

I am considering a geodesic γ∈H3(?1)γ∈H3(?1),
g=4∑i=1(dξi)21?∑i=1(ξi)2.
g=4∑i=1(dξi)21?∑i=1(ξi)2.
Now I want to make a small perturbation of the geodesic γγ and then get the curve γ?γ~, for any two ...

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### Reference request : Isomorphic stacks are given by Morita equivalent Lie groupoids

Let $\mathcal{G},\mathcal{H}$ be Lie groupoids. Let $B\mathcal{G}$ denote the stack of principal $\mathcal{G}$ bundles and $B\mathcal{H}$ denote the stack of principal $\mathcal{H}$ bundles. Then, we ...

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### The totally geodesic submanifolds of the 3-hyperbolic hypersurfaces do not intersect each other? [on hold]

I am considering a geodesic $\gamma \in \mathbb{H}^{3}(-1)$,
$$
g = \frac{4\sum_{i = 1}(d\xi^{i})^{2}}{1 - \sum_{i = 1}(\xi^{i})^{2}}.
$$
Now I want to make a small perturbation of the geodesic $\...

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157 views

### Reference request: Gauge theory [on hold]

What are some good introductory texts to gauge theory? I have some basic differential geometry knowledge, but I don’t know any algebraic geometry.
Also, as a side question, what intuitively is a ...

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### Formula for difference between curvature operators?

Let $(M,g)$ be a Riemannian manifold. Let $C:TM\to TM$ be symmetric positive definite. Define the metric
$$
(X,Y)_C = (X,CY)_g.
$$
Denote by $\nabla$ the Levi-Civita connection of $C$ and by $d^\nabla$...

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### Non-trivial foliation (excluding the Reeb foliation) [on hold]

Let $M$ be a closed oriented manifold, an oriented foliation $F$ is said non-trivial, if $F$ is not fibration of $M$, i.e. there does not exist a closed manifold $B$, such that $M\overset{F}{\to} B$.
...