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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Is there a Bishop-Gromov inequality for manifolds with boundary?
EDIT. Let $M^n$ be a smooth compact Riemannian manifold with smooth boundary.
Assume in addition that near the boundary $M$ is locally geodesically convex.
Assume that the Ricci curvature satisfies $...
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Geometry of the cut locus
Let $(M^n,g)$ be a smooth complete Riemannian manifold. Let $p\in M$ be a point. Recall that the cut locus of $p$ is the set of vectors $v$ in the tangent space $T_pM$ such that $\exp(t v)$ is a ...
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Is the space of Levi-Civita connections convex
More precisely, suppose we a given two metrics $g_0$ and $g_1$ on a manifold $M$. Let $\nabla_0$ and $\nabla_1$ be the corresponding Levi-Civita connections. Set $\nabla_t:=(1-t)\nabla_0+t\nabla_1$. ...
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heat kernel on closed manifolds - error in Chavel's book?
first of all, I am not sure if this question fits here. I asked this question on math.stackexchange also but didn't get an answer so far.
In Isaac Chavel's book Eigenvalues in Riemannian Geometry, ...
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Conditional Wiener measure continuous
consider a complete Riemannian manifold $M$ with heat kernel $p_M$ and let $U\subset M$ be an open set. Let $W_{x,t}^{y}$ be the Wiener measure associated to the Brownian motion starting at $x$ and ...
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construct divergence free vector field manifold
I have a smooth compact oriented manifold without boundary, $M$, with the volume form $\Omega$ and a Riemannian metric $g$.
Given a function $\phi$ is there any canonical way to obtain a divergence-...
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Bochner's formula on surfaces using moving coframes
I've read many times that moving coframes where a convenient tool for computations in Riemannian Geometry, especially on surfaces, but never really used it. Lately to get a better feel of the method, ...
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Curvature and intersection of submanifolds
Let $(M,g)$ be a Riemannian manifold of dimension $n$. (In the case I am interested in, $M$ is a complex symmetric domain, but I do not think that this is relevant for the question.)
Let $N$ be a ...
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Good references on the theory of harmonic mappings between Riemannian manifolds
I am currently reading the book by Professor Jost, "Riemannian geometry and geometric analysis", the last chapter on harmonic maps. It talks mainly about existence and regularity of the theory of ...
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Are metric isometries smooth at the boundary?
Let $M,N$ be smooth Riemannian manifolds with boundary (In particular, we assume the boundaries are smooth).
Suppose we have a map $\phi:M \to N$ which satisfies the following properties:
$$(1) \, \,...
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Open problems in sub-Riemannian geometry
What are some open problems in sub-Riemannian geometry?
I am interested especially in problems concerning connections and curvature, but any contribution is welcomed.
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Equivariant vector bundle structure on the tangent bundle of compact Riemann surfaces with non trivial action on the base space,
Let $M_{g}$ be the compact Riemann surface with $g\geq 2$.
Is there an infinit group $G$ with an equivariant action on the pair $(TM_{g}, M_{g})$ such that the action on the fibers preserves the inner ...
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optimal frame for a Riemann metric
In his excellent book, Harmonic maps, conservation laws and moving frames, Helein proves the existence of conformal coordinates on a surface, looking for an optimal frame. Let $\mathbb{D}$ equipped ...
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Existence of incomplete Riemannian metrics
During this question a manifold $M$ is meant to be a smooth connected (second countable, finite dimensional) manifold.
A Riemannian metric on a manifold $M$ is called complete if every geodesic is ...
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Gluing Alexandrov spaces along parts of boundary
I'm familiar with the Petrunin gluing theorem that states that gluing two Alexandrov spaces $M_1,M_2\in Alex(k)$ along their boundaries via an isometry $:\partial M_1\rightarrow \partial M_2$ results ...
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Some papers on the confluence of probability and geometry
I would like to get hold of some early papers on the confluence of Probability and Geometry. I am of course aware of the celebrated book, "Probability on Compact Lie Groups" but would like to ...
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Teichmüller space on non-orientable closed surfaces
It is known that any closed orientable surface of genus $g \geq 2$ admits a hyperbolic metric, and the Teichmüller space of such metrics has dimension $6g - 6$. I was wondering if there is a ...
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Uniformization for annuli with boundary
Let $(A,g)$ be a compact surface with boundary, diffeomorphic to the standard annulus $\{z\in\mathbb{C}:1\le|z|\le 2\}$, equipped with a smooth metric $g$.
Does there always exist a conformal (...
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Reference on these two affirmations on Differential Geometry
I'm reading the paper "A gap theorem for free boundary minimal surfaces in the three-ball".
In what follows $\Sigma$ is a minimal compact free boundary surface in the unit ball $B^3$ contained in $\...
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Ricci form is closed?
Let $(M,g,J)$ be an almost Kähler manifold and let $\rho$ denote its Ricci form
$$
\rho(X,Y) = \operatorname{ric}^{\mbox{c}}(JX,Y)
$$
where $\operatorname{ric}^{\mbox{c}}$ is the $J$-invariant part of ...
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Does a compact nonflat surface without conjugate points have ergodic geodesic flow?
I read this as a conjecture in the paper by Ballmann-Brin-Burns, titled "On Surfaces with No Conjugate points" JDG 25(249-273), 1987.
What is current status of this conjecture?
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When do geodesics reconverge?
Say I stand at the north pole and talk; in sufficiently frictionless conditions, one imagines that someone standing at the south pole could listen.
More generally, say $M$ is a compact Riemannian ...
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Covariance operator analogue for manifolds and respective measure manifolds
Assume $E$ is a connected riemannian manifold with geodesic metric space structure given by $d$ and $P$ is a probability measure over $E$ with Borel sigma-algebra given by this metric structure. Also ...
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Relation between flat and nilpotent structures on fibers?
When collapsing Riemannian manifolds under suitable curvature conditions, two types of structure arise on the fibers: flat structures and nilpotent structures. This depends on the scale at which one ...
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embedding in a Riemannian manifold
Let $M$ be a Riemannian manifold with boundary. Can it isometrically embed into a Riemannian manifold without boundary of the same dimension?
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Reference for homogeneous spaces
I am a graduate student of differential geometry.
I would like to get an overview over the way, how results are usually obtained for homogeneous spaces by Lie algebraic methods. By definition a ...
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Curvature of maximum of two riemannian metrics
Consider $g_1$ and $g_2$ two Riemannian metrics on a differentiable manifold $M$ of dimension $n\ge 4$. Suppose locally $g_i=f_i\sum_{j=1}^ndx_j^2$, where $f_i:M\rightarrow \mathbb{R}$ are non ...
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Reference request for structure equations
Let $(M,g)$ be a Riemannian manifold and let $\lbrace e_1,...,e_n\rbrace$ be a locally frame field on $M$ and $\omega _1 ,...,\omega _n$ be the dual $1$-forms of it. If $\omega _{ij}$ be the ...
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The comparison between the square of the functional value and the sum of squares of the $L^2$ norms of function and its Laplacian
I was reading a paper where I came across the following argument :
For any $x$ in $M$ and for a geodesic ball $B(x; \varepsilon)$ in a compact Riemannian
manifold $M$ with injectivity radius bigger ...
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Solutions of equations characterizing a complex structure
Let $(S^n,g)$ denote the unit $n$-sphere endowed with its induced metric $g$ from its embedding into $\mathbb{R}^{n+1}$. The Levi-Civita connection of $g$ induces a splitting of the tangent bundle of $...
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Manifold structure of the set of all smooth functions between two smooth manifolds!
In the studies of the calculus of variation, a map $f:M\to N$ said to be harmonic if it is a critical point of the Dirichlet energy function. i. e.
\begin{align}
E:C^\infty(M,N)&\longrightarrow \...
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Generalizations of Hopf-Rinow theorem
Let $(M,g)$ be a connected Riemannian manifold of dimension $n>1$. Then the Hopf-Rinow theorem states that $(M,g)$ is geodesically complete if and only if $(M,d_g)$ is complete as a metric space ($...
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Simply connected manifolds with dense geodesics on the tangent bundle
A unit speed geodesic $\gamma:I\to M$ on a Riemannian manifold $M$ can be lifted to a curve $\sigma=(\gamma,\gamma')$ on $SM$, the unit sphere bundle (the unit tangent bundle) of $M$.
Let us say that ...
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Least reasonable regularity on Riemannian metric tensor to define a metric [closed]
We assume that $M$ is a smooth manifold of dimension $m$, and we assume that $g$ is a smooth Riemannian metric $g$ on $M$.
If $M$ is moreover path-connected, then $g$ induces a metric (in the sense ...
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Why can we construct the following method?
I read in some papers the following method:
Let $h$ be a $(1,1)$-tensor field on 3-dimensional Riemannian manifold $(M,g)$ and $p\in M$. Then there exists a smooth local orthonormal basis of the ...
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Paneitz-Branson operator and Q-curvature
Let $(M,g)$ a n-dimensional Riemannian manifold, $n\neq 4$, and $h=u^{\frac{4}{n-4}}g$. Then the formula that connect the Panitz-Branson opertator $P_g$ and the Q-curvature $Q_{h}$ is
$Q_h=\frac{2}{...
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Evolution of $W_+$ and $W_-$ under the Ricci flow
In dimension $4$ the Weyl operator $W$ splits in two parts
$$W_+:\Lambda^{2}_{+} \to \Lambda^{2}_{+}$$
and
$$W_-:\Lambda^{2}_{-} \to \Lambda^{2}_{-}.$$
(a) Has there been a study of the evolution ...
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Splitting Short exact sequences of vector bundle with connection
Let $F\to M$ be a vector bundle and $E\subseteq F$ a subbundle. Suposse that $\nabla$ is a connection on $F$ s.t. preserves $E$, i.e. $\nabla_X(e)\in \Gamma E \quad \forall e\in \Gamma E, \ X\in\Gamma ...
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A Certain First-Order Differential Equation for a Closed 2-Form
Suppose we have a Riemannian manifold $(M,g)$ and a fixed vector field $X$. Consider the following equations for a 2-form $F$:
$$dF=0$$ $$(\delta-\iota_X) F=0$$
Here, $\delta$ is the codifferential i....
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Geodesics for non differentiable riemannian metric
Let $M$ be a differentiable manifold of dimension $n>2$ with a Riemannian metric $g=\sum_{i,j=1}^ng_{ij}dx_idx_j$ such that in some points on $M$ its coefficients $g_{ij}$ are not differentiable (...
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Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?
Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me:
Is there any open Ricci-flat ALE 4-manifold other than ...
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Einstein's field equation on orbifolds
I was wondering if there is some kind of Einstein's field equation for orbifolds (say semi-Riemannian of Lorentz signature if this make sense).
Here, by an orbifold I mean the "stacky" quotient of, ...
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Two ways a manifold can have little symmetry
Let $M$ be a closed connected smooth oriented manifold. The following two properties - that $M$ can either enjoy or not - intuitively both mean that $M$ has very little symmetry:
(a) Every self-map $...
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On geodesics of flat, non-symmetric connections
Let $M$ be a smooth $d$-dimensional manifold. Let $\nabla$ be a flat non-symmetric connection on $TM$ (i.e., curvature=0 and torsion$\ne$0). Let $p\in M$ and denote by $\exp_p:\Omega\subset T_pM\to M$ ...
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Does nonexpanding map between manifolds decrease volume?
(This question is a special case of a question I asked at SE, which got no answer there)
Let $M,N$ be diffeomorphic connected compact Riemannian manifolds, and let $f:M \to N$ be a surjective ...
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Frobenius norm of the principal submatrix of a uniformly distributed random orthonormal matrix
Suppose that we have a uniformly distributed $d\times d$ random orthonormal matrix $\mathbf{X}$. Here "uniform" is defined in the sense of Haar measure, i.e., the distribution does not change up to ...
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Clarification on Étienne Ghys' "Feuilletages riemanniens sur les variétés simplement connexes" paper
I apologize for this type of question, but I'm having some trouble to understand remark 3.4(4) on page 212 of this article, that reads
The restriction of $\overline{\mathcal{G}}$ (the foliation ...
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A variation of conformal vector field
Let $H$ be a smooth hypersurface in $\mathbb{R}^d$. I'm interested in vector field $X \in \mathfrak{X}(H)$ for which there is $f \in C^{\infty}(H)$ such that:
\begin{align}
\forall (u,v) \in \mathfrak{...
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sectional curvature with distance between two points
I am wondering if the following statement is true or not:
Let $ M $ be a complete Riemannian manifold with sectional curvature $ K $. Let $ \tilde{M} $ be the simply connected complete Riemannian ...
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Is a space with p-norm a Finsler manifold?
Suppose $\mathbb{R}^n$ is equipped with the p-norm $\left\Vert x \right\Vert_p$. Let $x\in \mathbb{R}^n$ and let $y$ be in a neighborhood of $x$. The distance between $x$ and $y$ can be defined as $\...
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