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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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      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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      95 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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      43 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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      130 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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      293 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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      234 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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      77 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
      285 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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      60 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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      1answer
      149 views

      Does $\nabla g=\omega(\cdot) g$ imply $\nabla$ is metric w.r.t a conformal rescaling of $g$?

      This is a cross-post. Let $E$ be a smooth vector bundle over a manifold $M$, where $\text{rank}(E) > 1,\dim M > 1$. Suppose that $E$ is equipped with a metric $g$ and an affine connection $\...
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      1answer
      140 views

      Is there exists (strictly) convex function on hemisphere?

      Given $\mathbb{S}^n_+:=\{x\in \mathbb{R}^{n+1}: |x|=1,x_{n+1}>0\}$ be the open domain in $\mathbb{S}^n$, or be viewed as the geodesic ball centered at the pole with radius $\frac{\pi}{2}$ in $\...
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      1answer
      215 views

      Can a hyperbolic manifold be a product?

      I was interested in whether a manifold which admits a metric of constant sectional curvature can be homotopy equivalent to a product of non-contractible manifolds. Of course, there are three cases: ...
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      100 views

      trapped geodesics

      Suppose $(M,g)$ is a compact Riemannian manifold with smooth boundary. We call a point $p \in M$ regular if there exists a geodesic of finite arc length passing through $p$ with end points on $\...

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