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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Christoffel symbols as the expansion coefficients of covariant/contravariant derivatives
Page 155 of Vector and Tensor Analysis with Applications, by A.I Borishenko and I.E. Tarapov, the authors state that Christoffel symbols of the second kind are expansions of $\frac{\partial {\bf e}_j}{...
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An application of Leray-Schauder degree theory for Nirenberg problem on the 2-sphere
I'm studying the article "The scalar curvature equation on 2- and 3-spheres" by Chang, Gursky and Yang and I'm particulary interested in the 2-sphere case.
They prove that if $K:S^2\...
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Arbitrary sectional curvatures at a point
Is it possible to choose a Lorentzian metric $g$ on a neighborhood of the origin in $\mathbb R^{1+n}$ so that the sectional curvature of all non-degenerate tangent timelike two planes at the origin is ...
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Diameter comparison for manifolds with $\text{Ric}\ge 0$
Let $M$ be a Riemannian manifold with $\text{Sec}\ge 0$. From Topogonov Theorem follows that for every $p \in M$ the quantity
$$
\frac{\text{Diam}(B_r(p))}{r}
$$
is non-increasing in $(0,\infty)$. ...
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What is a random eigenfunction on the hyperbolic plane?
Is there an (invariant under isometries) notion of a random eigenfunction on the hyperbolic plane, for a given eigenvalue?
It is a reference request because the answer is probably positive and I even ...
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What are the eigenvalues of the curvature operator on $\mathbb{C}P(2)$?
I tried asking this on stackexchange but was unsuccessful.
On page 150 of section 4.5.3 of Peter Petersen's Riemannian Geometry it is noted that, given an orthonormal basis $X,iX,Y,iY$ for $T_p\mathbb{...
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What are the tricks for computing/estimating Gromov-Hausdorff distance?
Gromov-Hausdorff distance (Wikipedia) between two compact manifolds measures how far away the manifolds are from being isometric.
In many cases it is possible to do coarse estimates and conclude that ...
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Classification of conformal diffeomorphisms of Minkowski space, part 2
This is a continuation of Classification of conformal diffeomorphisms of Minkowski space
Consider $\mathbb{R}^{n+1}$ equipped with the Minkowski (sign indefinite) metric:
$$g=(x^0)^2-(x^1)^2-\dots -(x^...
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Action of orientation-preserving isometric involution on complex structure
Let $(M, J, \omega)$ be a compact Kähler manifold. Let $\phi:M\to M$ is an orientation-preserving isometric involution.
Given a point $p\in M$ must there exist a decomposition $T_pM=\oplus_i W_i$ with ...
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Classification of similarity transformations of Minkowski space
Consider $\mathbb{R}^{n+1}$ equipped with the Minkowski (sign indefinite) metric:
$$g=(x^0)^2-(x^1)^2-\dots -(x^n)^2.$$
Is there a classification of diffeomorphisms $F\colon \mathbb{R}^{n+1}\tilde\to ...
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Non-isomorphic compact Kähler manifolds that are biholomorphic, symplectomorphic and isometric
Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$.
Assume there is a diffeomorphism $\nu:M\to N$ ...
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What is the relationship between Riemannian and sympletic musical isomorphisms on the cotangent bundle?
Let $M$ be a smooth manifold. Its cotangent bundle naturally has a symplectic structure, and this gives rise to musical isomorphisms. These musical isomorphisms are the ones from physics that relate ...
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Non-isomorphic compact Kähler manifolds not containing submanifolds biholomorphic to their conjugates
Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$.
Assume there is a diffeomorphism $\nu:M\to N$ ...
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Orientation-preserving isometric involution on compact Kähler manifold
Let $M$ be a compact Kähler manifold. If $\phi:M\to M$ is an orientation-preserving isometric involution does it have to be either holomorphic or anti-holomorphic?
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A Riemannian metric on $S^2 \times S^2$ of nonnegative curvature that is not a product
Good afternoon,
There is an example of a Riemannian metric on $S^2 \times S^2$ of nonnegative sectional curvature that is not a product metric. I know there is one; however, I cannot find a specific ...
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Non-symplectomorphic isometric compact Kähler manifolds
Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$.
Assume there is a diffeomorphism $\phi:M\to N$...
- 106k
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Numerical algorithms for geodesically convex optimization
I want to solve a minimization problem of the form
$\inf_{x \in M} f(x)$
where $M$ is a Hadamard manifold and $f$ is geodesically convex (but not differentiable). Since I know that in general a ...
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If the volume-ratio of an inscribed convex set to the circumscribing convex set is rational, can anything of consequence be further deduced?
Say, one has two $n$-dimensional convex sets $A$ and $B$, with $B$ being inscribed in the strictly larger set $A$. ($A$ and $B$ have at least one boundary point in common. $B$ “fits snugly” in $A$ ...
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References about Nomizu Conjecture
tI want to look for some references about Nomizu Conjecture(if $R(X,Y)R = 0$ Then $\nabla R = 0$),is there anyone know some papers/references about Nomizu Conjecture or the progress about Nomizu ...
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Geodesics and potential function
I try to assemble concepts of differential geometry for my own comprehension of the subject. I understand a manifold is a higher dimensional surface. It has a metric which perform inner product in the ...
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Spin-H structures
Let us define a Spin-H structure as a reduction of a SO(n)-bundle by the group: $$Spin^H (n)=Spin(n) \times SU(2)/\{ 1,-1\}$$ The Spin-H structures are analogous to the well-known Spin-C structures ...
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Uniformization theorem with boundary in the non-compact case
Let $\Sigma$ be a simply connected (and therefore orientable) smooth $2$-manifold with non-empty and connected boundary. Suppose that the interior $\operatorname{int}(\Sigma)$ is endowed with a ...
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How to study to learn differential geometry for applying it to statistics
Basically I want to learn information geometry or specifically the application of differential geometry in statistics to do a project. I am from a statistical background and have a knowledge about ...
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Spectrum of the Witten Laplacian on compact Riemannian manifolds
Below I have given what I am calling as the ${\rm Witten{-}Laplacian}_{s,p}$ on a Riemannian manifold $(M,g)$ for any constant $s >0$ and $p \in C^2(M,g)$
How generally is it true that this ${\rm ...
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Are invariant forms on homogeneous spaces necessarily closed?
Take a compact homogeneous space $G/K$, and a left $G$-invariant differential $k$-form $\omega \in \Omega^k(G/K)$. Will $\omega$ necessarily be closed? Might it even be harmonic when $G/K$ is endowed ...
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Geodesic line with endpoints in interior of Riemannian manifold or Alexandrov space
Let $X$ be a finite dimensional Alexandrov space with curvature bounded below and non-empty boundary. Let $\gamma$ be a shortest geodesic path in $X$ whose endpoints belong to the interior of $X$.
...
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Manifolds with boundary admitting no closed embedded minimal hypersurface
The following Theorem is proved in the paper entitled "Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex ...
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Is it possible to calculate the parallel transport on a loop from the Riemann curvature?
I admit I am not a differential geometer (a probabilist actually). However recently I get interested and I would like to have more intuitions and insight of what is the Riemann curvature.
This is the ...
- 106k
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Geometric interpretation of the Weyl tensor?
The Riemann curvature tensor ${R^a}_{bcd}$ has a direct geometric interpretation in terms of parallel transport around infinitesimal loops.
Question: Is there a similarly direct geometric ...
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Open neighbourhood of a point of space of Riemannian metrics
Let $M$ be a finite-dimensional compact smooth manifold and
$$\mathcal{M}et(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$
Q1-a: What metrics $g$ are very close to the given metric $g_0$? I.e....
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Extending Sobolev function on Riemannian manifold
Let $(M, \mu, d)$ be a geodesically complete non-compact Riemannian manifold such that measure $\mu$ is volume doubling, i.e. \begin{equation}\label{VD}\mu(B(x, 2r))\leq C\mu(B(x, r))\end{equation} ...
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What is the importance of singularities of type II in the Mean Curvature Flow?
I am reading the Mean Curvature Flow and Isoperimetric Inequalities by Manuel Ritoré, Carlo Sinestrari, Vicente Miquel and Joan Porti and I am curious to know what is the importance in understand the ...
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"Gross-Zagier" formulae outside of number theory
The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the ...
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An explicit (maybe algebraic) isometric embedding of the double torus with constant curvature K = -1
The following question is related to this previous question, Canonical immersion of the double torus:
Is there any known explicit (maybe algebraic) isometric embedding of a genus 2 surface endowed ...
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Canonical immersion of the double torus
It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the ...
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Concavity of the distance function to the boundary of Alexandrov space
I was told that the following fact is true.
Let $X$ be a finite dimensional Alexandrov space with non-negative curvature.
Then the function
$$x\mapsto dist(x, \partial X)$$
is concave (namely its ...
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Vector field along an immersion whose covariant derivative is the differential
Let $(M,g)$ be a Riemannnian manifold and let $f:\Sigma\to M$ be a smooth immersion. Then the vector bundle $f^\ast TM\to\Sigma$ has a natural bundle metric and metric-compatible connection. Can one ...
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Regularity properties of Minakshisundaram–Pleijel zeta function
Let $(M,g)$ be a closed (compact, no boundary) smooth $n$-dimensional Riemannian manifold. The Laplace–Beltrami operator $\Delta_g$ on $M$ has discrete spectrum $(\lambda_j)_j$ (indexed without ...
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Curvature collineation and the Killing identity
The Lie derivative of a general covariant $4$-tensor is given by
$$\mathcal{L}_{K}R_{abcd} = X^{e}\nabla_{e}R_{abcd} + R_{ebcd}\nabla_{a}X^{e} + R_{aecd}\nabla_{b}X^{e} + R_{abed}\nabla_{c}X^{e} + R_{...
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Concavity of distance to the boundary of Riemannian manifold
Let $(M^n,g)$ be a smooth Riemannian manifold with non-empty boundary $\partial M$. Assume (for simplicity) that $M$ is compact. Let $M$ be locally geodesically convex, i.e. any shortest path in $M$ ...
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When Riemannian manifold with boundary is Alexandrov space?
I am looking for a proof or, better, a reference to a proof of the following known fact.
Let $(M,g)$ be a smooth Riemannian manifold with boundary. Assume the sectional curvature of $M$ is at least $\...
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Shortest path on Riemannian manifold with boundary
Let $(M^n,g)$ be a smooth Riemannian manifold with non-empty boundary $\partial M$. Let $x\in \partial M$. Let $v\in T_x(\partial M)$ be a unit vector tangent to the boundary. Assume
$$II_{\partial M}(...
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How large can you draw an island on a map?
A cartographer friend asked me this question: could you classify (shapes of) islands by how much space they occupy on a map (comparatively to how much space is occupied by water) if you draw them as ...
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An abstract characterization of line integrals
Let $M$ be a smooth manifold (endowed with a Riemann structure, if useful). If $\omega \in \Omega^1 (M)$ is a smooth $1$-form and $c : [0,1] \to M$ is a smooth curve, one defines the line integral of $...
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A calculation involving Cotton tensor
I have a confusion regarding a calculation given below :
$$
\begin{split}
\int_M C^{ij}\nabla_i f \nabla_j f d\mu & = \frac{1}{3}\int_M g^{ij}g_{ij} C^{ij}\nabla_i f \nabla_j f d\mu \\
&= \...
- 178k
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mean curvature for codimension $>1$?
The mean curvature of a hypersurface in a Riemannian manifold is defined to be the trace of the second fundamental form. I was curious, does the notion of mean curvature generalise to higher ...
- 3,595
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Moving on Riemannian manifolds
Let $a,b,c\in\mathbb{R}^n$ such that $c$ is inside the $n$-disk with $a$ and $b$ as south and north poles. Then as $c$ moves toward $a$ through the line segment joining $a$ and $c$, $c$ is also moving ...
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Lorentzian cobordism through the dominant energy condition
Is the answer to the following problem, or some close variant thereof, known? Briefly:
Given two initial data sets $I_1=(M,g_1,k_1)$ and $I_2=(M,g_2,k_2)$, is there a time-oriented spacetime ...
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Product formula for Laplace de-Rham operator
Let $M$ be a Riemannian manifold with Laplace de-Rham operator $\Delta = (d + \delta)^2$. If $g$ is a smooth $k$-form, and $f$ is a smooth function, is there a simple formula for $\Delta(fg)$ when $k &...
- 6,735
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Curvature of complete conformal metrics on the open unit disk
Let $D$ be the unit disk in the complex plane, and assume that $g$ is a Riemannian metric on $D$ which is complete and conformal to the standard Euclidean metric. Can it be the case that the Gaussian ...
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