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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Homotopy bounds in simply connected complete Riemannian manifolds
Let $M$ be a simply connected complete Riemannian manifold, and let $x\in M$. Does there exist a nondecreasing function $R:\mathbb R_+\to\mathbb R_+$ such that, for every $r>0$ and all paths $\...
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Upper bound of derivative of exponential map
We know that for any simply connected surface $M$,whose Gaussian curvature $K\leq 0$, for any $p\in M$, $exp_p: T_pM\to M$ is diffeomorphism.
We know that for any $v\in T_pM$ and $w\in T_v(T_pM)\...
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Characterization of the Riemann curvature tensor [duplicate]
Let $(M^n,g)$ be a Riemannian manifold, $a\in M$ be a fixed point. It it well known that there exists a coordinate system near $a$ (e.g. the normal one) such that
$$g_{ij}(x)=\delta_{ij}+O(|x|^2).$$
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Are there compact Riemannian manifolds whith Q-curvature negative?
Are there known examples of compact Riemannian manifolds with Q-curvature negative?
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Some quantities which definitions are (somehow) similar to the classical Divergence
Motivated by classical formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$ and $L_{X} \Omega=Div(X) \Omega$ and the essential role of the diff operator $d$ in definition of divergence, we define some ...
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Obstruction to the existence of global isometries on a constant-curvature Riemannian manifold
Let $M$ be an $m$-dimensional simply connected Riemannian manifold that is not geodesically complete. Suppose $M$ has constant sectional curvature.
Because the curvature is constant, locally $M$ ...
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rigidity of eigenvalues of circular ensemble
Given a circular unitary ensemble, with the following joint density:
$p(\theta_1,\ldots, \theta_n) = Z_n \prod_{j < k} |e^{i \theta_j} - e^{i \theta_k}|^2$,
is the following statement true? With ...
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Do all surfaces (2d riemanian manifolds) admit constant curvature? [closed]
There seems to be a lot of theorems allowing to prove restricted cases of this (eg. uniformization, classification theorem for compact surfaces) . Intuitively, it seems true, but I've never seen a ...
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Geodesics on manifolds with boundary
Let $(M,g)$ be a Riemannian manifold with non-empty boundary. Is there any notion of injectivity radius on $(M,g)$ in points away from the boundary? By this I mean points lying in $M- \partial M$. ...
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Linearisation of Einstein operator
Let $(M,g)$ be a $(m+1)$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$.
The Ricci curvature can be viewed as a differential operator $\text{Rc}:\Gamma(S^2_+M)\rightarrow\...
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A geometric property of singular matrices
Let $S\subset M_{n}(\mathbb{R})$ be the singular points of the equation $Det=0$. That is $S$ is the critical points of the determinant function.
What matrices belongs to $S$, precisely?
Let $M=...
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The heat kernel as an exponential of an integral
In $\mathbb{R}^n$, if $\gamma$ is a line segment between $x_0 = \gamma (0)$ and $x = \gamma (t)$, one has the following formula:
$$\frac {\mathbb{e}^{- \frac{1}{4} \int_0^t <\dot{\gamma}, \dot{\...
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"Parallel translate" of a geodesic in the following sense [closed]
Since I'm lazy, I'm shamelessly referring to the following question:
Derivative of Exponential Map
Given a Riemannian manifold $M$, let $\gamma: (a,b) \to M$ be a geodesic and $E$ a parallel vector ...
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The Tangent Bundle of the Space of CR Structures on S^(2n+1)
Let $M$ be a smooth compact $n$-manifold without boundary, $g$ some choice of Riemannian metric on $M$, and $\omega_g$ the volume form gotten from $g$. Say you're interested in finding extrema for ...
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Does the green kernel converge as a series of functions?
Let $(M,g)$ be a compact rimannian manifold. It is well known that we can diagonalyse the Green kernel as a $L^2$ operator acting on functions. Moreover we have the convergence of the following series,...
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Besse p134 Riemann tensor in dimension 4
Does someone have a reference for the proof of 4.72 page 134 of Einstein Manifolds? It is said that
$$\check{R}-\vert R\vert^2g/4=S/3 (Ric-S/4) +2\mathring{W}(Ric -S/4) $$
because we are in dimension ...
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Have heat kernels for generalized Laplacians on non-compact manifolds been constructed?
Let $M$ be a non-compact Riemannian manifold which is "nice enough", and $D$ a generalized Laplacian on it. The construction of the heat kernel for the Laplace-Beltrami operator on $M$ seems to be ...
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Geodesics and Riemannian submersions
Let $X,Y$ be Riemannian manifolds, and $f\colon X\to Y$ be a Riemannian submersion.
Let $\gamma$ be a geodesic on $X$ starting at a point $x\in X$ and which is orthogonal to the fiber $f^{-1}(f(x))$.
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A question on the twistor space of a manifold
Let $M$ be either (a) self-dual conformal 4-manifold, or (b) hypercomplex $4n$-manifold.
In either case one can construct the twistor space $Z$ (in the case (b) $Z=\mathbb{C}\mathbb{P}^1\times M$ as a ...
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Coordinate charts on converging Riemann surfaces
Let $S$ be a $2-$dim manifold and $q \in S$. Furthermore, let $j_{n}$ be a sequence of complex structures on $S$ converging in $C^{\infty}_{\text{loc}}$ to a complex structure $j$ on $S$ as $n\...
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Parallel transport on a Hadamard manifold
Suppose, $X$ is a Hadamard manifold, i.e., a simply connected manifold of non-positive sectional curvature. Fix a point $w$ in $X$. Consider any three points $x, y, z$ in $X$. Let $\tau_{x, w}$ and $\...
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Certain construction of the Itô integral on manifolds
Let $M$ be a compact Riemannian manifold and let $X \in \mathfrak{X}(\mathbb{R}\times M)$ be a time-dependent vector field on $M$. I want to construct the Itô integral
$$ I(X) = \int_0^T \langle X(t, ...
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Compact riemannian manifolds with boundary that have infinite volume?
I am looking for references in the literature pertaining to (essentially riemannian) metric spaces that are compact of infinite volume, such in the following example. Consider a riemannian metric on ...
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Compact surfaces with boundary of constant negative curvature
Consider a surface (with boundary) diffeomorphic to $S^1 \times [0, 1]$ and with constant negative curvature, sitting inside $\mathbb{R}^3$. All the examples I know of such surfaces are "part of" (or "...
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Reference request: minimal (maximal) Lorentzian surfaces in $\mathbb{R}^{1,2}$
Let $R^{1,2}$ be the Minkowski 3-space, I would like to know any references about minimal (maximal) orientable Lorentzian surfaces in $\mathbb{R}^{1,2}$, including examples and maybe general theories, ...
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Distance function to a submanifold
Let $M$ be a compact Riemannian manifold and $\Sigma\subset M$ a closed submanifold. Given $x\in M$ we define the distance function to $\Sigma$ by $$d_\Sigma(x):=\inf\{d(x,y):y\in \Sigma\},$$ where $d$...
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Isotopy of positively curved surfaces of revolution in $\mathbb{R}^3$
Consider a surface of revolution of positive curvature. My question is, what are the surfaces (with boundary) in $\mathbb{R}^3$ which are isotopic to the surface of revolution, provided each member of ...
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Commutative spectral triples
The corresponence between compact Hausdorff topological spaces and commutative unital $C^*$-algebras is rather well known: Gelfand Najmark theorem gives perfect correspondence between these categories....
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Manifold with a quasi-positive curvature
As far as I know, in a simply connected compact manifold, still there exists no well-known obstruction for a manifold with a quasi-positive curvature to be a manifold with positive curvature.
But ...
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Relationship between Laplacian and Hessian on compact Lie groups
If $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is smooth and compactly supported, one has
$$\int |\Delta f(\mathbf{x})|^2\,d\mathbf{x} = \int \| Hf(\mathbf{x}) \|_F^2\,d\mathbf{x}\,,$$
where $\Delta$ ...
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Bounded functions dense in Sobolev Spaces
Let $M$ be a complete Riemannian manifold. Is it always true that the subspace $C^2_b(M)\cap W^{2,p}(M)$ is dense in $W^{2, p}(M)$, where $C^2_b(M)$ denotes the space of functions that are uniformly ...
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Isometries of Compact Semisimple Lie Groups
In this delightful question, the poster mentioned that the isometry group of a compact Lie group $G$, equipped with the metric from the Killing form, is $G\times G/Z(G)$, where $Z(G)$ is the center of ...
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Do geodesics in SL2R map to geodesics in the hyperbolic plane?
I am looking for a reference/proof/disproof of the following statement.
Equip the Lie group $SL_2(\mathbb{R})$ with the left-invariant Riemannian metric, whichis given on the Lie algebra by $\langle ...
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Dilatation of surface diffeomorphisms
Let $S$ be a higher genus surface, let $f\colon S\to S$ be a diffeomorphism and let $f_*\colon H_1(M)\to H_1(M)$ be the induced homology automorphism.
Define dilatation of $f_*$ as the largest ...
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Classification of spherical polygons
I need some sort of classification (up to isometry) of spherical polygons (i.e. polygons in $\mathbb{S}^2$ whose edges are given by geodesics) subject to the interior angles and the perimeter of the ...
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Continuous family of constant scalar curvature metrics
The question is as follows: Does there exist an example of a (continuous) family of metrics $g_t$ on a compact manifold such that the following properties hold?
All metrics $g_t$ have constant scalar ...
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Is the on-diagonal heat kernel “local” with respect to the metric?
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Let $X$ be a manifold, and $\mu_A$, $\mu_B$ two Riemannian metric on it which agree on an open subset $U\subset X$, i.e. $\mu_{A\,|U} = \mu_{B\,|U}$. Let $K_A(t;z,w)$ resp. $K_B(t;z,w)$ be ...
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Frobenius condition
Suppose X and Y are two unit length vector fields on a Riemannian manifold which are orthogonal at each point. Is it true that the lie bracket of X, Y belongs to the span of the vector fields at each ...
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Green's operator of elliptic differential operator
Let $P:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential operator, with index $=0$ and closed image of codimension $=1$, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...
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Sectional curvature as a Hamiltonian on the Grassmanization of the tangent bundle
Edit: According to the comments to the previous version of this question, I remove my essential errors in the question. I thank the commenters very much.
Let $M$ be a n dimensional manifold. ...
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Complex transport equation
Consider an n dimensional Riemannian manifold with boundary.
Let $\Phi$ be a complex valued smooth function defined in M. Does there exist a NONE VANISHING complex valued function $u$ that solves the ...
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Surfaces with specific types of second fundamental form
Given a three dimensional Riemannian manifold $(M,g)$ and a surface $\Sigma \subset M$ can one categorize surfaces where the second fundamental form of $\Sigma$ is a scalar multiple of the induced ...
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New setting for tensor definition
This is the standard definition for a tensor in a smooth manifold (Nakahara, for example):
But while reading Salamon's Riemannian Geometry and Holonomy Groups I have found this rather different ...
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Frobenius Condition for a specific first order pde
I would appreciate it if Someone would be kind enough to share some insights about the following question:
Suppose $(M,g)$ is a 3 dimensional Riemannian manifold. Consider the following system of ...
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metric on ${\bf SPD}_n({\mathbb R})$
The cone ${\bf SPD}_n({\mathbb R})$ of symmetric positive definite matrices is endowed with a nice geometrical structure. The midpoint of the (unique) geodesic between $A$ and $B$ is the so-called ...
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Formula for the Perimeter of a spherical triangle?
Consider the ordinary sphere $\mathbb{S}^2\subset \mathbb{R}^3$ and a spherical triangle $T\subset \mathbb{S}^2.$ I'm looking for a formula from which the perimeter $P$ of $T$ is "computable" given ...
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Isothermal-related functions in higher dimensions
I am interested in getting some geometrical or analytical perspective in studing the following complex pde. I would appreciate any help.
Consider $ (M,g)$ to be a 3 dimensional Riemannian manifold ...
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Minimal surfaces + Semi-Geodesic Coordinates
Let $(M,g)$ be a three dimensional smooth Riemannian manifold and suppose that $\Gamma$ is an embedded minimal surface in $M$. Define the Fermi or semigeodesic coordinates around this surface through ...
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Obstructions for existence of a Riemannian metric such that a given function is harmonic
Let $f:\mathbb{R}^{n}\to \mathbb{R}$ be a smooth function. What type of obstructions exist for existence of a Riemannian metric $g$ on $\mathbb{R}^{n}$ such that $f$ is a harmonic ...
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Handle body of 3-manifold with boundary
We know from Morse theory that smooth manifold(with or without boundary) is a handlebody.
However, I found a paper "Three-dimensional manifolds with boundary of nonnegative Ricci curvature" by Ananov, ...
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