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        Questions tagged [riemannian-geometry]

        Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.

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        Domain of exponential map

        Let $M$ be a Riemannian manifold and $exp_p:T_{p}M\rightarrow M$ be the exponential map. Let $\gamma_v$ be a geodesic starting at $p$ with the $\gamma_v'(0)=v$. Also define $I_v:= Domain (\gamma_v)$ ...
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        151 views

        Isometric embedding of a genus g surface

        Can a genus $g$ surface with constant negative curvature and $g>1$ be isometrically embedded in $\mathbb{R}^4?$
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        49 views

        Ricci flow preserves locally symmetry along the flow

        Let $(M,g_0)$ be a closed locally symmetric Riemannian manifold and let $g(t)_{t\in[0,T)}$ be a solution to the Ricci flow on $M$ with $g(0)=g_0$. How one can prove that Ricci flow preserves locally ...
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        Feynman-Kac formula for domains with boundary

        As far as I know (I am not an expert), any solution of the heat equation on $\mathbb{R}^n$ or on a closed Riemannian manifold with the given initial condition can be presented in terms of stochastic ...
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        47 views

        A particular semi-linear equation on Riemannian manifolds

        Let $m\in \mathbb{N}\setminus \{1\}$ and suppose $(M,g)$ denotes a compact smooth Riemannian manifold with smooth boundary and consider the semi-linear equation $$-\Delta_g u+q(x)u + a(x)u^m=0\quad \...
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        97 views

        Example of a manifold with positive isotropic curvature but possibly negative Ricci curvature

        Is there any example of a manifold with a positive isotropic curvature but it possibly obtains a negative Ricci curvature at some point and the direction? If we see the definition of the positive ...
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        46 views

        Does the Volume Ratio of a Geodesic Ball for a Complete Riemannian Manifold tend to the volume of a Unit Ball in Euclidean $n$-space?

        I am reading Peter Topping's notes on Ricci flow: on page 99 a statement is made which is needed for his proof of a version of Perelman's no local volume collapse theorem, but I am not sure why it ...
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        135 views

        More general form of Fourier inversion formula

        My question begins as follows: Suppose $G$ is a compact (Lie) group and $A$ is a $G$-module, and denote the action of $G$ on $A$ by $\alpha$. Fix $a\in A$ and view $$ f:g\mapsto \alpha_g(a) $$ as an $...
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        2answers
        300 views

        Is this a submanifold?

        Let $(M,g)$ be a compact Riemannian manifold with an isometric action $\rho : G \to \mathrm{Iso}(M)$ by a compact Lie group $G$. There is a natural extension of $\rho$ to $TM$ given by: $$\psi : G \...
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        1answer
        241 views

        Laplacian spectrum asymptotics in neck stretching

        Let $M$ be a compact Riemannian manifold. Let $S \subset M$ be a smooth hypersurface separating $M$ into two components. Let $g_T$ be a family of Riemannian metric obtained by stretching along $S$, i....
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        1answer
        92 views

        Hyperkähler ALE $4$-manifolds

        It is well known that Kronheimer classified all hyperk?hler ALE $4$-manifolds. In particular, any such manifold must be diffeomorphic to the minimal resolution of $\mathbb C^2/\Gamma$ for a finite ...
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        78 views

        A quantity associated to a compact Riemannian manifold with boundary(The pair of Laplacian)

        Let $M$ be a Riemannian manifold with boundary $\partial M$. Are the following operators, Fredholm operators?Is there a geometric terminology and geometric interpretation for the fredholm index of ...
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        2answers
        290 views

        Volume of manifolds embedded in $\mathbb{R}^n$

        Let $N$ be a closed, connected, oriented hypersurface of $\mathbb{R}^n$. Such a manifold inherits a volume form from the usual volume from on $\mathbb{R}^n$ and has an associated volume given by ...
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        50 views

        G-invariant Kazdan--Warner problem

        Kazdan and Warner at the $70$'s published a serie of papers on the problem of ensuring the existence of metrics with prescribed scalar curvature. For instance: https://projecteuclid.org/euclid.bams/...
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        62 views

        Off-diagonal estimates for Poisson kernels on manifolds

        Let $(M,g)$ be a complete Riemannian manifold, $\Delta$ its Laplace-Beltrami operator and $T_t = (e^{t \Delta})_{t \geq 0}$ the associated heat semigroup. We can define the subordinated Poisson ...

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