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        Questions tagged [dg.differential-geometry]

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

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        49 views

        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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        votes
        0answers
        55 views

        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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        votes
        0answers
        52 views

        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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        0answers
        148 views

        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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        votes
        0answers
        73 views

        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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        votes
        1answer
        130 views

        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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        vote
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        58 views

        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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        1answer
        49 views

        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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        30 views

        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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        0answers
        52 views

        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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        117 views

        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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        votes
        1answer
        156 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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        0answers
        101 views

        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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        1answer
        78 views

        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 $M\overset{F}{\to} B$. ...
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        votes
        0answers
        31 views

        Adiabatic limit of the torus bundle on the circle

        Let $(S^1,g_1)$ be a circle with length $L$ and $(T^2,g_2)$ a flat torus where $T^2=\mathbb C/\{\mathbb Z \oplus \mathbb Z \tau\}$ for $\text{Im}\, \tau>0$. If $(M^3,g_{\epsilon})$ has a fibration ...

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