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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Alexandrov-Fenchel theorem for Riemannian manifolds [closed]
Let $E,M$ be two convex subsets of $\mathbb{R}^n$. Then the Alexandrov-Fenchel theorem states that the coefficients $V_j(E,M)$ in
$$\operatorname{Vol}(xE+yM)=\sum_j {n\choose j} V_j(E,M)x^{n-j}y^j$$ ...
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Curve length in the Sasaki metric
I am trying to read Appendix II.A.2 (Distances in the tangent bundle) in Canary, Epstein, Marden (eds.), Fundamentals of Hyperbolic Manifolds: Selected Expositions and am stumbling over a calculation ...
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Unbounded curvature implies infinite diameter on complete metric spaces
I recently asked this question Unbounded sectional curvature implies infinite diameter?.
I would like now to ask something similar, but in another context.
Suppose you have a complete metric space $(...
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What is known about warped product metrics satisfying conditions more general than conformal flatness?
In this paper, the authors characterize warped product metrics which are conformally flat (the fibers must have constant sectional curvature, on some cases there is a limitation on the number of ...
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Calculation of the top Chern class of spinor bundle over $S^{2n}$
It's well known that for a complex vector bundle $E$, we have
$$c_n(E)=e_n(E_\mathbb{R}) $$
But I'm very curious about the relationship between the top Chern class of spinor bundle and the Euler class ...
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Is the oriented bordism ring generated by homogeneous spaces?
I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ...
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Going from piecewise to genuine geodesic without decreasing number of intersections?
Let $(M^2,g)$ be a complete, two-dimensional Riemannian manifold be given; also given is $\gamma: [0,\infty) \to M$, an injective geodesic in $M$.
Suppose there are two geodesic segments $\gamma_i : [...
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Largest hyperbolic disk embeddable in Euclidean 3-space?
Hilbert proved that there's no complete regular ($C^k$ for sufficiently large $k$) isometric embedding of the hyperbolic plane into $\mathbb{R}^3$. On the other hand, the pseudosphere is locally ...
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Trace-class heat semigroups
Let $(M,g)$ be a compact Riemannian manifold and $\Delta_g$ its Laplace operator.
Let $\varphi$ be a test function on $\mathbf{R}_{>0}$. We define the operator on, say, $L^2(M)$
$$T_{\varphi}(u) :=...
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Gravitational instantons metric (change variables)
I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric:
$$
\gamma dz d\bar{z}+\gamma^{-...
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About "residual" scalar curvature in Einstein warped product manifold
I have an Einstein warped product manifold $M=B \times_f F$ with warped metric $g=g^B+f^2g^F$, where $F$ is Ricci flat, then its scalar curvature is $S^F=0$.
It is well known that the scalar curvature ...
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Planar sections of convex sets in Cartan-Hadamard manifolds
Let $X$ be a convex set in Euclidean space $\mathbf{R}^n$ and $p\in\mathbf{R}^n$ be a fixed point. Then any plane $\Pi$ passing through $p$ intersects $X$ in a convex set. Conversely, this property ...
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In which ways did geometric flows and variational methods from Riemannian geometry enter the symplectic world?
I am interested to learn about the role of geometric analytic methods for solving problems in symplectic geometry, In particular, I would like to know what results heavily rely on this machinery (incl....
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What is the shape of the perfect coffee cup for heat retention assuming coffee is being drunk at a constant rate?
Note: I asked this on Mathematics SE and even though @TheSimpliFire offered a bounty on it, no-one had a good answer
Find the optimal shape of a coffee cup for heat retention. Assuming
A constant ...
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The tangent map of the exponential map
Let $(M,g)$ be a Riemannian manifold and let $\exp^M:TM\to M$ denote the exponential map. Its tangent map $T\exp^M$ is a map $TTM\to TM$. The connection on $TM$ induces a canonical metric on $TM$, the ...
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Closure of Laplacian
Let $(M,g)$ be a complete Riemannian manifold and $\Delta$ the (positive) Laplace-Beltrami operator. Now, consider this operator as an operator
$$\Delta:\mathcal{D}(\Delta)\to L^{2}(M)$$
There are two ...
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Roadmap to learning about Ricci Flow?
Hello,
I'm curious to what books etc. one could use to understand the basics of Ricci flow, what areas of math are needed and so? What areas should one specialize in? See it as a roadmap to ...
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Smooth isometric embedding of euclidean n-space into an arbitrarily small neighborhood of another euclidean space
It is not hard to isometrically embed the euclidean plane smoothly into an arbitrarily small neighborhood of euclidean 4-space: First embed a real line isometrically in an arbitrarily small ...
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Domain of spectral fractional Laplacian
Let $(M,g)$ be a complete Riemannian manifold with Laplacian $\Delta:C^{\infty}_{c}(M)\to C^{\infty}_{c}(M)$ (think of $\mathbb{R}^{d}$ if you wish). This operator is essentially self-adjoint in $L^{2}...
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Finite subgroups of $\mathrm{O}_n(\mathbb R)$ from finite subgroups of $\mathrm{SO}_n(\mathbb R)$
Let $G$ be a finite subgroup of $\mathrm{SO}_n(\mathbb R)$. We also assume $G$ to be "maximal" in the sense that for every $g\in\mathrm{SO}_n(\mathbb R)\setminus G$, we have that $\overline{\...
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Existence of adjoint operators on manifolds
Let $(M,g)$ be an oriented Riemannian manifold and $V$ a finite-rank vector bundle equipped with a non-degenerate bundle metric $\langle\cdot,\cdot\rangle_{V}$. This bundle metric, in turn, gives rise ...
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Foliation of spaces
It turns out that a very important idea to derive properties for a bigger space is to try to foliate the space, derive the same property for each leaf and patch everything up to get the desired ...
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Moser iteration in dimension $6$
Let $M$ be a closed Riemannian manifold of dimension $6$. We have a function $f\geq 0$ on $M$ satisfying
\begin{align*}
\Delta f \leq gf-\frac{3}{4}f^2
\end{align*}
Where $g$ is another smooth ...
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If a triangle can be displaced without distortion, must the surface have constant curvature?
Suppose $S$ is a Riemannian 2-manifold (e.g. a surface in $\mathbb{R}^3$).
Let $T$ be a geodesic triangle on $S$: a triangle whose edges are geodesics.
If $T$ can be moved around arbitrarily on $S$ ...
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Curvature of curves through a point of a surface smoothly embedded in Euclidean space
The curve C(𝜃) drawn on a smoothly embedded surface 𝜮 in 3-space — where C(𝜃) is defined as the intersection of 𝜮 with a 2-plane perpendicular to 𝜮 at P — leaving the point P at angle 𝜃 will ...
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Sufficient condition for existence of a closest-point projection from a neighborhood onto a subset of a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold and let $N$ be a subset of $M$.
On one hand, it is well known that if $N$ is an embedded submanifold of $M$, then it admits a tubular neighborhood, and, ...
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What is the least dimension of the Euclidean space into which every Riemannian flat n-torus embeds isometrically?
Given an integer $n > 0$, let $f(n)$ denote the least dimension of the Euclidean space into which there exists a C∞ isometric embedding of every Riemannian flat $n$-dimensional torus $\Bbb T^n =\...
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Volume comparison with integral curvature bounds which gives lower bound on volume on Riemannian manifold
In Riemannian geometry, the Bishop-Gromov theorem states that for an $n$-dimensional manifold $M$ with nonnegative Ricci curvature $Rc\geq 0$, then the volume quotient
$$
\frac{\text{Vol} B(x,r)}{\...
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Self-adjointness of fractional laplacian
Lets consider the following functional analytic definition of the fractional Laplacian: Consider a (complete, connected, oriented) Riemannian manifold $(M,g)$ with corresponding Laplacian $\Delta_{g}$....
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Is there any physics theory which is similar to these analogies?
Since I am doing this little "research" project on my spare time and in my physical neighborhood there are not many people to discuss these ideas, I wanted to share with you a small point of ...
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What does $\nabla^i f$ mean?
I am reading the article Some Geometric Calculations on Wasserstein Spaces of John Lott and there is this covariant index in the covariant derivative: $\nabla^i$. And I don't quite understand it.
In ...
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Gradient descent under the presence of symmetries
Let $M$ be a Riemannian manifold (I'm happy to assume it is Euclidean space) with a function $f: M \to \mathbb R$ and a group of isometries $G$ acting on $M$ and preserving $f$, i.e., $f(gm) = f(m)$ ...
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Question about interpretation of algebraic notation in differential geometry paper
I am unable to understand the notation of equations (1.1) and (1.6) in page 2 of Kowalski and Belger's paper "Riemannian metric with the prescribed curvature tensor and all its covariant derivatives ...
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Does the curvature determine the metric?
I ask myself, whether the curvature determines the metric.
Concretely: Given a compact manifold $M$, are there two metrics $g_1$ and $g_2$, which are not everywhere flat, such that they are
not ...
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A specific question in $G_2$ geometry
Let's take a closed $G_2$ manifold $(M,\Phi)$. $\Phi$ denoting the three-form which defines the $G_2$ structure on $M$. Let's take a closed two form $\theta\in\Omega^2(M).$ Is
\begin{align}
d(\theta_{\...
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Dynamical obstruction for a vector field to have a Harmonic divergence
Let $(M,g)$ be an analytic Riemannian manifold and $X$ be an analytic vector field on $M$. Can we always have a volume form $\Omega$ such that $\operatorname{Div}_{\Omega} X$ is a harmonic ...
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Correct curvature tensor of symmetric space of positive definite matrices with trace metric?
Let $Pos(n)$ be the set of $n \times n$ real positive definite matrices with trace (aka affine-invariant) metric
$$\langle u, v \rangle_p = tr(p^{-1} u p^{-1} v)$$
for all $p \in Pos(n)$ and $u, v \in ...
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A (possible) Lie algebra extension of the Lie algebra of a foliation
Motivation: The aim of this post is to extend the Lie algebra of a foliation to a bigger Lie algebra. We assume that a manifold $M$ is foliated by compat leaves. The Lie algebra of the foliation is ...
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Invariants associated to a principal bundle whose total space is a symplectic manifold acted symplectically by group structure
The following question - proposal came to my mind about 4 years ago but I did not find any solution to this question and did not find any answer via e-personal comunication with some ...
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An identity for the higher form Levi-Civita connection
Take $M$ a Riemannian manifold and $\Lambda^1$ its space of one forms. The LCC (Levi-Civita connection) $\nabla:\Lambda^1 \to \Lambda^1 \otimes \Lambda^1$ is well known to satisfy the identity $m \...
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When is the Chern integral given by the norm of the curvature tensor?
I saw somewhere that for a Kahler manifold that admits a Kahler-Einstein metric the following integral formula is true.
$$\int_M c_2 \wedge \omega^{n-2} = \frac{1}{n(n-1)}\int |Rm|^2 \omega^n$$
It ...
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Lamination as limit of arcs
I am reading Bonahon's notes on closed curves, in particular the part about hyperbolic laminations. In his notes Bonahon illustrates some examples as why laminations should be "limit curves" on ...
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volume of exceptional group orbits
Assume that $G$ is a compact group acting by isometries on a (compact) Riemannian manifold (M,g), with principal orbits of dimension $d>0$. For $x\in M$, let $G(x)$ denote the $G$-orbit of $x$, by $...
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Understand Riemannian cross-derivative on product manifolds
Suppose we have a smooth function $f:\mathcal{M}\times\mathcal{N}\rightarrow\mathbb{R}$ where the domain is a product of two Riemannian manifolds. The Riemannian cross-derivative ([1], section 2) is ...
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Differentiability of distance function at first conjugate point
Formulation of question: Consider a complete, simply connected Riemannian manifold $M$ with Riemannian metric $d$. For $x,y \in M$ that are distinct but close enough to each other, there is a geodesic ...
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How to construct a $C^1$ variation of the unique, minimal geodesic at its first conjugate point
Motivation for the question: This is a question that I encountered when reading Proposition 1.8 of the paper "Optimal transport and curvature" by Alessio Figalli and Cedric Villani, whose ...
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Convergence of spectrum
Let $M$ be a compact manifold and $g_k$ be a sequence of Riemannain metrics smoothly converging to another Riemannian metric $g$.
Let $\{\lambda^k_j\}$ be the spectrum of the Laplacian of the ...
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Lee-Parker Yamabe problem proposition 4.6
I believe there may be a gap towards the end of the proof of proposition 4.6 in the Bulletin of the AMS paper The Yamabe Problem by Lee and Parker : https://projecteuclid.org/journals/bulletin-of-the-...
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Exponential convergence of Ricci flow
I've been trying to understand the asymptotic behavior of Ricci flow, and there are two facts which I am unable to square away. I'm interested in higher dimensional manifolds, but my question is ...
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What's the relationship between the Riemannian metric and Jacobi field?
I encounter to the question in reading the following Excise:
Let $(M,g)$ be a $m$-dimensional Riemannian manifold, and $(r,\theta^1,\theta^2,\dotsc,\theta^{m-1})$ be the (geodesic) polar coordinate. ...
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