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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Conjugate points in Lie groups with left-invariant metrics
For any Lie group $G$ there exist many left-invariant Riemannian metrics, namely, one just takes any inner product on the tangent space at the identity $T_eG$ and then left translate it to the other ...
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Equality of the determinants of certain submatrices of an orthogonal matrix
Is the determinant of any submatrix of an ORTHOGONAL matrix extracted from the intersection of $k$ row and $k$ columns equal to that of the $(n-k)(n-k)$ submatrix remaining after deletion of these ...
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Negative curvature in the middle of $R^{3}$
What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside?
Basically, I am asking for a ...
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Random metrics on compact orientable surfaces
Hello everyone,
Let $S_g$ be a compact orientable surface of genus $g \geq 2$, and let $\mathcal{A}$ be the set of $\mathcal{C}^{\infty}$ Riemanniann metric on $S_g$ endowed with the topology of ...
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Why don't $\mathbb{T}^n, \mathbb{S}^n, \mathbb{H}^n$ admit other metrics of constant curvature?
The torus $\mathbb{T}^n$, the sphere $\mathbb{S}^n$ and the hyperbolic space $\mathbb{H}^n$ admit metrics of constant (sectional) curvature $0, 1, -1$ respectively. Do they afford metrics of constant ...
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Geometry defined by foliation.
In $\mathbb R^3$ there are 3 natural foliations given by the lines parallel to each axis, which intersect transversally. Let $M^n$ a manifold with $n$ foliations by lines or circles that intersect ...
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Basic results in bounded geometry
I'm doing analysis (dynamical systems) in the context of Riemannian manifolds of bounded geometry and I find myself reproving quite a few standard results/tools from standard differential geometry, ...
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Surjectivity of the normal exponential map
Given an isometric (in the Riemannian way) immersion $f:N\rightarrow M$ between complete, smooth riemannian manifolds, are there conditions on $M$, $N$, $f$, such that the normal exponential map $\...
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Special coordinates for periodic metrics
This question is a follow-up to that one.
Given a $\mathbb{Z}^n$-periodic metric $g$ on $\mathbb{R}^n$ (with $n>2$), is it possible to find a periodic diffeomorphism $\varphi$ such that $\varphi^*...
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Volume growth of covers and growth of deck-transformation groups
It is well-known that if $\widetilde M\to M$ is a Galois cover of a compact Riemannian manifold $M$ with deck-transformation group $G$, then the growth of $G$ equals the volume growth of $\widetilde M$...
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Reference request: Riemannian manifold of linear isometries from $\mathbb{C}^n$ into $\mathbb{C}^m$
Does anyone know a citeable reference which works out the properties (geodesics, geodesic distance, ect) of the Riemannian manifold of linear isometries from $\mathbb{C}^n$ into $\mathbb{C}^m$, $m>...
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Can one (block) diagonalize the curvature matrix of 2 forms on a Riemannian manifold?
Let $M$ be a smooth Riemannian manifold, let $R$ be the Riemannian curvature operator, and let $p$ be a point in the manifold. With respect to any orthonormal basis of the tangent bundle at the point $...
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recognizing Kahler manifolds of complex dimension n
Is there new classification of Kahler manifolds of complex dimension n and new results for necessary and sufficient conditions for a manifold being Kahler? I know if redactivity of Lie algebra on ...
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Invariants of a $GL(3,\mathbb{R})$ action
I'm trying to understand the standard $GL(3,\mathbb{R})$ action on the 15-dimensional space of possible values for the derivative of the Riemann curvature tensor of a 3-dimensional manifold $M$ at a ...
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A riemannian manifold with finitely many closed contractible geodesics
By a closed geodesic, I mean a smooth periodic geodesic $\mathbb{R} \rightarrow (M,g)$. I will consider them up to geometric distinction.
This means that any two closed geodesics are equivalent if ...
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Positively curved metrics on $S^2\times S^2$
As you know, the Hopf conjecture is about the existence of positively curved metric on $S^2\times S^2$. Hsiang-Kleiner have shown that there exists no positively curved metric admitting $S^1$-action ...
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On Dimension of Instanton Moduli Space
I am reading Charles Nash's book on differential topology and QFT. In particular, I have question on the part calculating dimension of instanton moduli space. The question split into conceptual part ...
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Riemannian manifolds with small geodesics and bounded curvature
Let $(M,g)$ be a compact riemannian manifold with sectional curvature $|K_g| \leq 1$. A lemma due to Klingenberg asserts that then either the injectivity radius $i_g \geq \pi$ or $(M,g)$ contains a ...
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Energy functional
During my study on Ricci Flow I faced some functional known as energy functional. For example Einstein-Hilbert functional is called an energy functional, also in Perelman's works $\mathcal{F}(g,f)=\...
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Is geodesic plane field a Killing field?
Let $M$ be a closed orientable Riemannian manifold. Recall that a plane field on a Riemannian manifold is said to be geodesic if any geodesic tangent to the plane field at one point is tangent to it ...
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Homogeneous Spaces and Equivariant Hodge Maps
For a homogeneous space $G/H$, endowed with a $H$-equivariant metric $g$, let $\ast$ be the corresponding Hodge star map. It seems that $\ast$ must also be $\ast$-equivariant, but I can't see how one ...
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Strongly parabolic PDE vs weakly parabolic PDE
In my studies on the Ricci flow, I was faced with a problem. To prove the existence and uniqueness of solutions to the Ricci flow, it is proved that the Ricci flow is a Parabolic PDE type. Then one ...
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geometric meaning of Ricci-flatness
What is the geometric meaning of Ricci-flatness? We know that if the Riemann tensor at a point vanished, manifold is flat at this point. but I don't know When the Ricci tensor vanished at a point, ...
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Is there a lower bound for variance in terms of curvature?
If the Gaussian curvature of the metric $g= f^2(x,y)(dx^2+dy^2)$ is nonzero then $f$ cannot be constant. This can be expressed by stating that the (probabilistic) variance $Var(f)$ of $f$ is nonzero (...
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3-dim 1-connected Alexandrov manifold with curvature $\ge 0$ Heomomorphic to sphere?
For Alexandrov manifold in the title we mean 3-dim Alexandrov apace which is also a topological. manifold.
Shioya-Yamaguchi posted a conjecture on their paper "Collapsing 3-manifold with lower ...
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Trivial canonical bundle of a Ricci-flat, simplyconnected Kähler manifold
Hallo,
I have two questions where I do not really know how to deal with them. Let $(M,J,g)$ be a Kähler manifold, where $g$ is the Riemannian metric and denote by $\omega(\cdot , \cdot) = g(J \cdot ,...
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Geometry of Hopf fibrations and the fibration of Steifel Manfiolds over Grassmannians
When $F = \mathbb{R}, \mathbb{C}$ or $\mathbb{H}$, there are fibrations $$O(k,F)\rightarrow V_k(F^n)\rightarrow G_k(F^n)$$ where $V_k(F^n)$ are Steifel manifolds and $G_k(F^n)$ are Grassmannians. When ...
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The Origin of the Musical Isomorphisms
In Riemannian geometry, the "lowering indices" operator is denoted by $\flat:TM \to T^*M$ and the "raising indices" operator by $\sharp:T^*M \to TM$. These isomorphisms are ...
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A k-form is thought of as measuring the flux through an infinitesimal k-parallelepiped
On the wikipedia has written "A $k$-form is thought of as measuring the flux through an infinitesimal $k$-parallelepiped." How does a $k$-form do this? if this sentence is right, then the flux of ...
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Resolvent of Laplacian
Hello!
Let $(M,g)$ be a Riemannian manifold and $-\Delta$ the Laplacian on M (acting on smooth functions). Then the resolvent $R(\xi)$ of $-\Delta$ is a compact operator.
Is it possible to find for ...
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Bi-invariant Riemannian metrics on So(n)
Defining the inner product - 1/2 tr(XY) on the lie algebra so(n)(skew symmetric matrices) is one way to introduce a bi-invariant metric on So(n) since the inner product is ad-invariant. Are there any ...
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Is every topological (resp. Lie-) group the isometrygroup of a metric space (resp. Riemannian manifold)?
The isometry group of a metric space is a topological group (with the compact open topology). The isometry group of a Riemann Manifold is a Liegroup. (Thm. of Steenrod-Myers)
So, is every topological ...
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Different notions of geodesics.
Let $M$ be a (without boundary and not necessarly complete) Riemannian manifold.
A map $c\colon [a,b]\rightarrow M$ is called geodesic of type A iff $c$ is piecewise smooth, parametrized proportional ...
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positive sectional curvature of submanifold in $R^n$?
Let $N$ be a hypersurface in $\mathbb R^n$, assume it is compact. Then the maximum point of $d(O, x)$ when restrict to $N$ has positive sectional curvature lower bound by the one of the correspond ...
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Clarification in a paper
This is regarding a clarification in page 384 of a paper published in Annals of Statistics by Amari.
In page no. 384, he defines $$R_i(t)=\frac{\partial}{\partial \theta_i} D_{\alpha}(q(x,t),p(x,\...
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Normal coordinates
On a Riemannian Manifold $(M,g)$, for any $p \in M$, there exists a normal coordinate system at p, ${U, x_1, ..., x_n}$. I want to integrate over a normal coordinate system with U being a geodesic ...
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Approximating solutions to minima of the discrete Lagrangian
I have been stuck on this problem for a week and I'm not sure whether or not it is hard or I'm just missing something obvious.
General gist of the problem
I have a variational problem on a Riemannian ...
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Diameter estimate of distance sphere of positive curved manifold
Let $M$ be an $n$-dimensional Riemannian manifold with sectional curvature lower bound 1. Fix a point say $O\in M$, let $S(r)$ denote the distance sphere centered at $O$ with radius $r$. The classical ...
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Reference - Asymptotic geodesics on compact surfaces without conjugate points
I would like to ask about possible references on the following problem: consider a compact surface and a metric without conjugate points. Consider it's universal covering endowed whith the lifting of ...
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The shape operator and an almost contact structure of a real hypersurface in $\mathbb{C}^n$
Let $S$ be an immersed real hypersurface in the Euclidean $\mathbb{C}^n$ with the standard complex structure $J$. Let $A:T(S)\rightarrow T(S)$ be the shape operator of $S$ (e.g. w.r.t. the outer ...
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Induced Riemannian metric on Jet-Manifold
Suppose $(M,g)$ and $(N,g')$ are smooth Riemannian manifolds and $J^r(M,N)$ is the
smooth manifold of $r$-jets $j^r_xf$ of smooth maps $f:M\to N$.
Is there an 'induced' Riemannian metric $g''$ on $J^...
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$C^k$ topology of metrics
Is the space of Riemannian metrics, over a compact manifold, complete when endowed with the $C^k$-topology of metrics?.
Is there a good reference for this?
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Hessian of the inverse exponential map on a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold. Then define
$f: T^*M \times M \to \mathbb{R}$
$f(x,\xi, y) = \langle exp_x^{-1} y, \xi \rangle$
where $exp_{\cdot}\cdot$ is the the exponential map and it's ...
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Characterizing Hessians among symmetric bilinear tensors
I apologize in advance if this is somewhat elementary, but:
Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\in Sym^2(M)$ ...
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Lie derivative of curvature
Let $M$ be a Kahler manifold, with Kahler metric $g$. Let $X$ be a holomorphic Killing vector field of $g$, i.e. $L_{X} g = 0$, where $L_{X}$ is the Lie derivative along $X$. Let $R$ be the Riemannian ...
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Why is it important that partial derivatives commute?
I am asking this in the context of differential geometry (specifically Riemannian).
When the Levi-Civita Connection is defined, we require that the torsion tensor is 0, which in local coordinates ...
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Compact surface with genus$\geq 2$ with Killing field
Let M be a compact Riemannian surface of genus$\geq 2$.
Can M have a globally defined Killing field ?
Can M have a Killing field defined on M-(finite set of points)?
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Tangent space, metrics etc. on simplicial sets
Is there a way to attach some sensible notion of tangent space to a simplicial set? If yes, is it possible to transfer typical local data from differential geometry such as metrics to this setting?
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When a Riemannian manifold is of Hessian Typ
When a Riemannian manifold is of Hessian Type (i.e., a Riemannian manifold which its metric is Hessian)
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Positivity of second fundamental form implies global convexity?
Let $M$ be a Riemannian manifold of dimension $n$. Let $N\subset M$ be a subset with smooth boundary $\Sigma=\partial N$. If one assume the second fundamental form $II$ with respect to inner normal ...
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