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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Are there some intrinsic invariants of surfaces other than Gaussian curvature?
The principal curvatures of a surface is denoted by $\kappa_{1}, \kappa_{2}$.
Let $P(x,y)$ be a polynomial with real coefficients. Assume that $P(\kappa_{1}, \kappa_{2})$ is an intrinsically ...
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If $f$ is dynamically coherent, is there a unique invariant foliation tangent to $E^{c}$?
Let $f$ be a diffeomorphism of a closed manifold $M$ such that $f$ is partially hyperbolic if the tanget bundle of $M$, $TM$ splits into three invariant sub-bunbles
$$
TM = E^{s} \oplus E^{c} \oplus ...
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Yamaguchi submersion theorem
Let me remind first a theorem of Yamaguchi (1996).
Given $n\in \mathbb{N}, \mu_0>0$. Then there exist $\delta_n>0$ and $\epsilon_n(\mu_0)>0$ with the following property. Let $X$ be an $n$-...
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Differentiability of geodesics in Alexandrov subspaces of Riemannian manifolds
Let $M$ be a smooth Riemannian manifold. Let $X\subset M$ be a closed path connected subset which has curvature bounded below in the sense of Alexandrov with respect to the induced intrinsic metric. ...
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Normal variation of embedded surfaces [closed]
Let $M^3$ be a complete riemannian manifold and $\Sigma ^2\subset M^3$ a embedded minimal compact surface. Consider the normal variation $\phi: \Sigma \times \Bbb{R}\to M$ given by
$$\phi(p,t)=\exp_p(...
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(quasi)metric on Riemannian manifolds via Brownian Motion?
Given points $a$ and $b$ on a Riemannian manifold $M$, I would like a (quasi)metric that corresponds to some property of Brownian Motion from $a$ to $b$ (or rather, to $\epsilon$-ball $B = \{ x : |x - ...
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Splitting of totally geodesic Riemannian foliations
Let $\mathcal F$ be a non-singular Riemannian foliation on $(M,g)$ whose leaves are totally geodesic. Suppose further that the leaves are Riemannian products of irreducible manifolds $L=L_0\times ...\...
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geodesics on $G/K$ which are not the orbits of a 1-parameter subgroup of $G$
Let $G$ be Lie group and $K \subset G$ a closed subgroup, such that there exists a $v \in T(G/K)$ whose isotropy-group $G_v$ is discrete (so iff $\dim G_v =0$). Lets assume $g$ acts properly on $T(G/K)...
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Smoothing of the distance function on a Riemannian manifold
Suppose $(M,g)$ is a complete Riemannian manifold. $p\in M$ is a fixed point. $d_{p}(X)$ is the distance function defined by $p$ on M (i.e., $d_p(x)$=the distance between $p$ and $x$). Let $\epsilon&...
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Geometrically-explicit upper bound for on-diagonal heat kernel
Let $M$ be a compact Riemannian manifold, and $K(t;z,w)$ the heat kernel associated to the usual Laplace-Beltrami operator on functions. There are results of the form
$$K(t;z,z) \leq \frac{C_M}{f_z(t)...
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Unexpected regularity of the distance from a $C^2$ submanifold
Let $\Gamma$ be a $C^2$ compact submanifold of $\mathbb{R}^n$. Consider the distance function $\delta$ from $\Gamma$. It is well known that, for sufficiently small $\varepsilon>0$, $\delta$ is $C^2$...
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Two questions about Li-Yau-Hamilton estimate
This question is from my question on mathematics.
Picture below is from 231 page . For to prove $Q\ge 0$ on $M \times (0,T)$,
$(\partial_t -\Delta)Q \ge 0$ and $Q\ge 0$ are needed to prove.But I ...
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If $X$ is a compact smooth Riemannian manifold, why don't we integrate on a fundamental domain in the universal cover? [closed]
Let $X$ be a compact connected Riemannian manifold. The metric gives a local volume form. The universal cover is orientable, and has a precompact subspace locally isometric (with the covering metric) ...
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Uniform approximation of a continuous flow by a $\mathcal{C}^1$ flow
Setup: Consider a (smooth) compact Riemannian manifold $M$, whose distance is denoted by $d$. Let $\Phi$ be a continuous flow, namely a continuous application from $\mathbb{R} \times M $ to $M$ ...
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Horizontal distribution of a totally geodesic foliation
Let $\mathbb{M}$ be a $n+m$ dimensional manifold. Consider on $\mathbb{M}$ a rank $n$ sub-bundle $\mathcal{H}$ of the tangent bundle. We assume that $\mathcal{H}$ is endowed with a fiber wise inner ...
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Definition of Levi-Civita connection map and a theorem about it?
Does anyone know definition of Levi-Civita connection map that defined as $K: TTM\to TM$. and how to prove the following theorem:
Theorem: If $X\in\mathfrak{X}(M)$ be a vector field over $M$ and $K:...
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Applications of Hessian operator in the Riemann manifold. Simple samples $S_{2}(f)$
Study article R. C. Reilly is entitled Applications of Hessian operator in the Riemann manifold had a doubt in the remark, shortly after the theorem 2 of that Article.
The theorem is stated as: ...
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Limit of argmin of sum
Suppose that I know $f_n\rightarrow f$ and $g_n\rightarrow g$ are both continuous maps from a Complete Riemmanian Manifold $X$ to $\mathbb{R}$ which converge pointwise almost everywhere. Then is it ...
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General Relativity and Differential Geometry intuitions of Second Bianchi Identity
In General Relativity, one uses the Riemann Tensor in its coordinate form $R_{abcd}$, and proves the Second Bianchi Identity-
$R_{abcd;e} + R_{abde;c} + R_{abec;d} = 0$
It is said that ...
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Counterexample to volume comparison inequality assuming only scalar curvature bound?
The Gromov-Bishop volume comparison theorem says that if we have a lower bound for the Ricci curvature on $(M,g)$, then its geodesic ball has volume not greater than the geodesic ball with the same ...
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Convergence of Riemannian metrics spectra
Consider a one-parameter real analytic family of metrics $g_t$ on a compact manifold $M$ converging to a metric $g$ in $C^k$-norm, for some $k$. It is known that the Laplace spectrum of $g_t$ will ...
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Gromov-Hausdorff limits of 2-dimensional Riemannian surfaces
Let $\{M_i\}$ be a sequence of 2-dimensional orientable closed surfaces of genus $g$ with smooth Riemannian metrics with the Gauss curvature at least $-1$ and diameter at most $D$. By the Gromov ...
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Dimension of curvature invariants
EDIT: Let $V$ be a Euclidean space and let $O(V)$ denotes its orthogonal group.
Let $K(V)\subset Sym^2(\wedge^2(V))$ denote the subspace of curvature tensors, i.e. the subspace of elements satisfying ...
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References for metrics in matrix groups
I am studying a very concrete matrix group with a riemaniann (right invariant) metric for solving a question on Applied Math. I need explicit formulas for the distance between two matrices, geodesics ...
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Converse to Lichnerowicz Vanishing Theorem?
The Lichnerowicz vanishing theorem says that if on a compact 4-dimensional spin manifold there exists a metric whose scalar curvature $R>0$, then there are no harmonic spinors; $$D\psi=0 \implies \...
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On the curvature tensor with certain conditions
Let $(M^{n+m},g)$ be a Riemannian manifold and let $\lbrace X_1,...,X_n,Y_1,...,Y_m\rbrace $ be a locally orthonormal frame for $M$($3\leq n,m$).
If we suppose the curvature tensor $R$ of $g$ ...
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Index of Modified Dirac Operator
Let's say we have an oriented compact 4-d Riemannian spin manifold $(M,g)$. Everybody who's anybody has heard about the index of the Dirac operator $D: S^+\rightarrow S-$; it's the $\hat{A}$-genus, ...
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Harmonicity on semisimple groups
I asked this on Math.SE and got no answer, so I'll try my luck here.
Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in ...
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Compact manifolds locally bi-Lipschitz to Euclidean space
I have a compact manifold $M$, and I am allowed to choose some Riemannian metric on it, exactly which I don't care. But I would love it if I could choose the metric $g$ such that every point has an ...
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Tameness in $\mathbb{R}^{n^2}$ of the subset consisting of matrices of positive determinant
The Lie group $GL(n)$ being a manifold is locally path-connected. Consider its connected component of the identity $C\subseteq\mathbb{R}^{n^2}$. What is a good way of showing that $C$ is a tame ...
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Is there a smooth manifold which admits only rigid metrics?
Does there exist a (finite dimensional) smooth manifold $M$, such that every Riemannian metric on $M$ has no isometries except the identity?
Of course, such a manifold must not admit a diffeomorphism ...
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How isometric action on Riemannian manifold acts on cut locus
Assume that $M$ is a simply connected closed Riemannian manifold with no boundary and nonnegative sectional curvaure Assume that ${\bf Z}_n=(g),\ n\geq 3$ acts on $M$ isometrically. Then if $gx=x$, i....
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Zariski open set in orthogonal grassmanian [closed]
I am confused about the following question.
Consider $\mathbb C^4$ endowed with nondegenerate symmetric bilinear form $J:=\left(\begin{matrix}0&0&0&1\\0&0&1&0\\0&1&0&...
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Is Fano Kahler surface with reverse orientation also Kahler?
In particular, do Fano Kahler surfaces with reverse orientation admit Kahler-Einstein metrics?
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Harmonic function with injective boundary conditions is an immersion?
Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are given an immersion $f:M \to \mathbb{R}^n$. (i.e $df$ is invertible at every point $p \in M$, note ...
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When is a Riemannian metric equivalent to the flat metric on $\mathbb R^n$?
I'm looking for an easily-checked, local condition on an $n$-dimensional Riemannian manifold to determine whether small neighborhoods are isometric to neighborhoods in $\mathbb R^n$. For example, for ...
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isometric action on the $n$-sphere
Let $S^n$ be the $n$-sphere. If $n=2k+1$ is odd, then we can identify $S^n$ as a subset of $\mathbb{C}^{k+1}$. We define the $S^1$ action on $S^n$ by multiplication, namely
$$ \Psi \colon S^1 \times ...
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coisotropic action on $TS^{2n+1}$
Let $S^{2n+1}$ be the $m$-dimensional sphere in $\mathbb{C}^{n+1}$. Endow $S^{2n+1}$ with the standard metric. Let $S^1$ act by multiplication on $S^{2n+1}$. Then $S^1$ and the canonical action of $SU(...
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Characterizing left invariant and right-$O_n$ invariant distances on $GL_n$
Consider the group $GL_n(\mathbb{R})$ with its standard topology.
It is not hard to show that there exists Riemannian metrics on it which are left-$GL_n$ and right-$O_n$ invariant. (In fact it's ...
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A problem of defining addition in a Quotient space
Let $\mathcal{C}$ be the space of all parametric curves $x:[0,1]\rightarrow \mathbb{R}^2$. Let the set of all re-parameterizations of curves is $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma (...
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co-dimension one minimizing verifolds
It is known that a minimizing co-dimension verifolds within a manifold may need to be singular. I think a famous example first partially analyzed by Jim Simons is the cone on in the 8-ball of the ...
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Is $\mathbb{P}T^*M$ a sub-Riemannian manifold if $M$ is Riemannian?
(this question is about a particular aspect of a previous question, which was not duly stressed)
Let $(M,g)$ a Riemannian $n$-dimensional manifold, and let
$$
\widetilde{M}:=\mathbb{P}T^*M
$$
be the $...
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Willmore functional
Let $(M^2,g)$ and $(\bar{M},\bar{g})$ be two Riemannian manifolds. Suppose that $\mathcal{W}$ is the Willmore functional on the set of immersion functions from $M^2$ to $\bar{M}$.
We know that $(\...
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Distance comparison in submanifold versus in the underlying manifold
Let $(M,g)$ be the (underlying) manifold, $(S,g|)$ be a submanifold. Let $a,b,c \in S$. It's not in general true that $d_M(a,b)\leq d_M(a,c) \implies d_S(a,b)\leq d_S(a,c)$.
QUESTION I:
The above ...
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When are the minimizing geodesics of a totally geodesic submanifold also minimizing in the underlying manifold? [duplicate]
Also asked here: https://math.stackexchange.com/questions/1725787/when-are-the-minimizing-geodesics-of-a-totally-geodesic-submanifold-also-minimiz
A reference on totally geodesic submanifold (TGS):
...
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Applying Cheeger and Colding segment inequality
The question turns out quite long and maybe a bit vague, I apologize in advance for that.
I am currently trying to understand Cheeger and Colding proof of the almost splitting theorem. Currently I am ...
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Criterion for deciding the conformal class of a metric on a complete surface
For orientable closed Riemannian surfaces $(S,g)$, there is a constant curvature metric $\overline{g}$ on $S$ that is conformal to $g$ in the sense that $\overline{g} = e^ug$ for some smooth function $...
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Different complexifications of a real analytic Riemannian manifold
I have a question concerning the complexification of a real analytic Riemannian manifold. Let $(M,g)$ be a compact Riemannian manifold. It is a well known fact that in a neighbourhood $U$ of the zero ...
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On the complexification of a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle. If we suppose $TM\otimes\mathbb{C}$ is the complexification of $TM$ then how can we define a natural metric on the complex bundle $...
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Finding a specific Global Smooth Function
Any help with this problem would be appreciated. Thanks
Suppose $(M^3,g)$ is a smooth compact Riemannian manifold with smooth boundary and $\gamma$ is a simple smooth orientable curve in $M$. Does ...
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