# 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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### $L^2$ norm of scalar curvature

As mentioned by Wilie Wong, I modified to the following verison:
Let $M$ be a closed smooth $4$ manifold.
Q
Suppose that $c>0$ is any positive number, can we find a Riemannian metric $g$ on $M$, ...

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### The convex hull of a manifold whose cobordism class is trivial

Let $M$ be a compact orientable $n$ dimensional manifold. Assume that $M$ has trivial cobordism class.
Is there an embedding of $M$ in some Euclidean space $\mathbb{R}^m$ such that the convex ...

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### Closed almost geodesics in a Riemannian manifold

Let $M$ be a smooth Riemanniann manifold. For $\varepsilon \geq 0$ we call an $\varepsilon$-geodesic (I am not sure that this is a standard name) a smooth map
$$\gamma\colon [a,b]\to M$$
such that for ...

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### Good resources that talk about geodesically convex sets for riemannian manifolds?

Are there any good resources that talk about geodesically convex sets, and in particular convex hulls, for riemannian manifolds? I’ve been wanting to learn more about them, their geometry and ...

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### Explicit computation of the vertical and horizontal vector bundles

Given a closed Riemannian manifold $(X,g)$ and let $p\colon TX\to X$ be the usual projection, the paper I'm reading asserts that the Levi-Civita connection induces a splitting $T(TX)= H(TX)\oplus V(TX)...

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### Is the volume functional analytic in the space of embeddings? What about locally?

Let $(M^{n+1},g)$ be an analytic Riemannian manifold and let $\Sigma^n$ be a closed analytic manifold. Denote by $\operatorname{Emb}(\Sigma, M)$ the space of all smooth (or maybe analytic) two-sided ...

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### Cross product of two infinitesimal bendings

Let $M$ be a smooth (embedded or immersed) surface in $\mathbb{R}^3$.
Let $Z_1,Z_2$ be two vector fields along $M$, thought of as $\mathbb{R}^3$-valued functions, satisfying the following differential ...

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### Does the first Laplacian eigenfunction on a homogeneous space have a unique maximum?

For convex domains $\Omega \subset \mathbb{R}^n$ with Dirichlet boundary conditions, it's known that any first Laplacian eigenfunction is log-concave. In particular, it has a unique maximum.
These are ...

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### Self-ajointness of the Laplacian over a Riemannian manifold with boundary

I have some doubts about on a passage found on this article (https://arxiv.org/pdf/1510.08136.pdf).
Let
$(M,g)$ be a Riemannian manifold with boundary;
$E\to M$ be an hermitian fiber bundle;
$\Delta$ ...

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### Question about deformation of the metirc on a Riemannian manifold

I'm a bit confused with the deformation of the metric on a given Riemannian manifold $(M,g)$ with a smooth boundary. How can we deform the metric $g$ such that it is a product near $\partial M$, ...

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### Geometric intuition behind definition of $\delta$-necklike points of the Ricci flow

In "The Ricci Flow: An Introduction", the authors define a $\delta$-necklike point of the Ricci flow as a point $(x, t)$ where $$\|\text{Rm} - R (\theta \otimes \theta)\| \leq \delta \|\text{...

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### Convergence of extremal subsets in Alexandrov spaces

Let $\{X_i^n\}$ be a sequence of $n$-dimensional Alexandrov spaces with curvature uniformly bounded from below which converges in the Gromov-Hausdorff sense to a compact $n$-dimensional Alexandrov ...

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### Classic Hölder spaces versus Hölder-Zygmund spaces on Riemannian Manifolds

Let $M$ be a compact Riemannian manifold without boundary.
The spaces ${C}^k(M)$ are defined as usual for $k \in \mathbb{N}$ and we can define Hölder spaces ${C}^s(M)$ $s \geq 0$, $s \notin \mathbb{N}$...

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### Action of fractional Laplacian on Hölder / Besov spaces on Riemannian manifolds

Let $M$ be a compact Riemannian manifold (without boundary) and $\Delta$ be the corresponding (positive) Laplace-Beltrami operator. We also define the operators $I^s = (\mathrm{Id} + \Delta)^{-s}$ for ...

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### Norm of a $(1, 1)$ form on a Kähler manifold

Given a Kähler manifold $(M, g)$ what is the convention for defining the inner product on two $(1,1)$ forms $\alpha = \sqrt{-1}\alpha_{i \bar k} dz_{i} \wedge d \bar{z}_{k}$ and $\beta = \sqrt{-1}\...

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### Manifold with totally geodesic boundary is spin if and only if its double is spin

Let $(M,g)$ be a Riemannian manifold with totally geodesic boundary $\partial M$. Let $(DM,Dg)$ be the double of $(M,g)$ obtained by reflection of across $\partial M$. I'm looking for a reference for ...

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### Prescribing variations that preserve the area

Let $(M^3,g)$ be a Riemannian manifold and let $\varphi : \Sigma \to M$ be a two-sided embedding of a closed surface into $M$, with a unit normal $N$. Suppose that $\varphi$ is a regular point of the ...

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### Connection between a function and its usage in geometry [closed]

I know nothing about geometry, but I found a function which seems to have something to do with geometry.
This function is, $$f(x,y,z) = \dfrac{(x,y,z)}{\sqrt{1 + x^2 + y^2 + z^2}}$$
where $x,y,z$ is ...

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### What does the boundary of convex hulls look like in matrix Lie groups?

Let $G$ be a compact matrix Lie group under the Killing form metric $\langle \xi, \eta \rangle_g = -\frac{1}{2}\text{tr}((g^{-1}\xi)^T(g^{-1}\eta))$ for $g \in G$ and $\xi,\eta \in T_gG$. Let $C \...

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### Area of a deformation of a closed surface

Let $(M^3,g)$ be a complete Riemannian manifold. Fix a two-sided immersion $\varphi : \Sigma^2 \to M$ from a closed surface into $M$, with unit normal $N$. Given $f \in C^\infty(\Sigma)$ and $\alpha : ...

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### Changing the system of PDE by diffeomorphism

This problem comes from the book Hamilton's Ricci flow.
Given a smooth functional $f$, and following system. $$\partial_tg_{ij}=-2(R_{ij}+\nabla_i\nabla_jf)$$ If there exist a 1 parameter family of ...

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### Harmonic functions on complete Riemannian manifolds

I have started reading a paper of Colding and Minicozzi, where they prove that on a complete Riemannian manifold $M$ of non-negative Ricci curvature, the space of harmonic functions of growth order at ...

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### A problem arising from reading a lecture on the Yamabe problem of how the Hölder inequality is used

I'm reading Tawfik - The Yamabe problem: the PDE is
$$
\Delta \varphi+h(x) \varphi=\lambda f(x) \varphi^{q-1}. \label{1}\tag{1}
$$
Theorem (Yamabe). For $2<q<N=N=2 n /(n-2)$, there exists a $C^{\...

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### Decomposition of tensors

It is well known that every traceless, symmetric $2$-tensor can be decomposed uniquely into a Lie derivative part and a Codazzi part. Is there an analog for totally symmetric $k$-tensors?

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### Question about Clifford volume element

I'm a little confused with the following:
Let $M$ be a $m$ dimensional Riemannian manifold and $e_1,\cdots,e_m$ be a local orthonormal base of $TM$. Let
$$
\omega_\mathbb{R}=c(e_1)\cdots c(e_m)
$$
...

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### Decomposition about splitting of symmetric spaces of compact type

I get stuck in the following question:
Why does a locally symmetric space of compact type $M$ split locally irreducible components of dimension $\geq 2$ which are Einstein? In particular, why are all ...

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### Questions about symmetric spaces

I'm a little confused with the following questions:
(1) Why does a symmetric space $M=G/K$ of compact type have $\mathcal{R}^M\geq 0$?
(2) Moreover, why does $\chi(M)\neq 0$ if and only if ${\rm rk}\ ...

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### Why are we interested in spectral gaps for Laplacian operators

Let $M$ be a Riemannian manifold and let $\Delta$ be its Laplacian operator. There is a large literature on a spectral gap for such a $\Delta$, that is, finding an interval $(0,c)$ which does not ...

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### Quasilinear second order parabolic equation

For the following parabolic equation
\begin{equation*}
\begin{split}
u_t&=\frac{u_{xx}}{1+u_x^2}-\frac{1}{u}\\
&u(x,0)=\cosh x+1.
\end{split}
\end{equation*}
How to show that $u(0,t)\sim \sqrt{...

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### Manifolds with nonpositive radial curvature

How can one construct examples of Riemannian manifolds which have nonpositive radial curvature about some point, but are not nonpositively curved everywhere? (I presume that they exist, but do not ...

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### Finding an asymptotically flat manifold with ${\rm Ric}_{r\phi} = \frac{\sin\theta}{r^2}$

Let $(r,\theta,\phi)$ be the spherical coordinates on $\mathbb{R}^3$ where $\theta \in (0,\pi)$ and $\phi\in (0,2\pi)$.
Does there exist an asymptotically flat metric $g$ on $\mathbb{R}^3\setminus B_1$...

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### Question about spin map

I'm confused with the following definition of a spin map.
A spin map is a map $f: N\to M$ between differentiable manifolds such that their second Stiefel-Whitney classes are related $\omega_2(N)=f^*\...

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### Homogeneous metrics on compact Lie groups

Given a differentiable manifold $M$, a Riemannian metric $g$ on $M$ is called homogeneous if its isometry group acts transitively on $M$. In that case, given any group $G$ acting transtitively on $(M,...

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### Weitzenböck formula and comparison of norms

Let $M$ be a closed Riemannian manifold with a spin$^\mathbb{C}$ bundle $S$. Now for a spin connection $A,$ and a spinor $\phi,$ it can be shown that $C\lvert\nabla_A\phi\rvert^2\geq \lvert D_A\phi\...

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### Manifolds and Riemannian geometry with a bundle viewpoint

I was wondering if there are any books that builds the theory on manifolds and Riemannian geometry, but at the same time treats these subjects in the general case of bundles (similar to Jeffery Lee's ...

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### Existence of a local spinor bundle

I am confused about the existence of a local spinor bundle.
My question is that if a Riemannian manifold $M$ is not spin, why does there exist a local spinor bundle over all sufficiently small open ...

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### Hodge theory in higher eigen-spaces?

Hodge theory for elliptic complexes $E$ identifies the space of harmonic sections with cohomology
$$\mathcal{H}(E) \simeq H(E).$$
A classical example with differential forms ($E = (\Omega,d)$) ...

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### Non-negatively curved manifolds and the volume of balls

Whether a complete non-compact non-flat Riemannian $n$-manifold $M$ with non-negative sectional curvature has Euclidean volume growth?
That is, whether there is a constant $C>0$ such that $\mathrm{...

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### Question about Dirac operator

Let $D$ be a generalized Dirac operator on a complete Riemannian manifold. I'm a little confused to prove that there exists a constant $c>0$ such that
$$\|D\sigma\|^2\geq c^2\|\sigma\|^2$$
for $\...

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### What are the volume-preserving diffeomorphisms of hyperbolic space?

What are the volume-preserving diffeomorphisms of $d$-dimensional hyperbolic space (in say the hyperboloid model)?
In particular, I'm especially interested in: what are the volume-preserving ...

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### Mapping torus of Klein bottle

This got 5 upvotes but no answers on MSE (Mapping torus of Klein bottle), so I'm cross-posting to MO:
The mapping torus of a Klein bottle $ K $ is a compact flat 3 manifold.
The mapping class group of ...

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### Is it known whether a closed simply-connected manifold of non-negative curvature admits positive Ricci?

It is discussed in this question whether a simply-connected closed Riemannian manifold with non-negative Ricci curvature admits positive Ricci curvature, and the answer appears to be "no, there ...

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### How negative Hessian of Hamiltonian implies strict convexity?

Let $X$ be a strictly convex domain in a Riemannian manifold $(\tilde{X}, g)$ of dimension $\geq 3$ with boundary defining function $\rho$ (so $\rho \in C^{\infty}(\tilde{X}), \rho>0$ in $X,<0$ ...

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### conformal flat near a point

Let $(M,g)$ be a Riemannian manifold, and $x\in M$ be a fixed point.
Q Can we find a conformal transformation such that near $x$ we can write $e^{2u}g$ as $(dx^1)^2+\cdots+(dx^n)^2$?
Since the ...

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### On the convergence of the spectral decomposition of a harmonic function

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$...

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### Relation between the distance projective maps and their angles

Let $f:N \to \mathbb{R}^2$
be a differentiable map of smooth manifolds. Let $\mathbb{R}^2$ be decomposed as a direct sum of line bundles, i.e. $\mathbb{R}^2=E(x) \oplus F(x)$, where $F(x)$ and $E(x)$ ...

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### Bubble tree convergence: Why is it necessary to consider centre of mass of the energy measure?

In the paper “Bubble Tree Convergence for Harmonic Maps” by Thomas H. Parker, after the picking the energy concentration points, he proceeded by expanding the map around each energy concentration ...

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### Is there a geometric intepretation of the trace of tensor on a Riemannian manifold?

For a long time I thought the trace for matrices were just an elementary function with nice properties. But it is much more than that and really should be think of as a geometrical object (see ...

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### Exponential map on the Hilbert manifold $\Omega(M,q_0,q_1)$

Consider $M$ to be a compact manifold and consider the based loop space $\Omega(M,q_0,q_1)$ of loops of class $W^{1,2}$. It can be shown that that this has a structure of an hilbert manifold. It's ...

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### Application of Santalo’s formula

Suppose that $(M,g)$ is a compact smooth Riemannian manifold with a smooth boundary and suppose that $f$ is a smooth function on $M$ with the property that
$$ \int_I f(\gamma(t))\,dt=0,$$
for any ...