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.
2,971
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Reference for parallel transport around loop and its relation to curvature
It is a well known fact that the geometric meaning of a linear connection's curvature can be realized as the measure of a change in a fiber element as it is parallel transported along a closed loop.
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singular integral operators
Let $(\Omega,g)$ be a compact domain with smooth boundary and suppose that $g$ is smooth. Let $g_D$ and $g_N$ denote the Dirichlet and Neumann green functions for the Laplace-Beltrami operator.
My ...
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A Lagrangian connection and its algebraic interpretation
Assume that $M$ is a $n$ dimensional Riemannian manifold. So $TM$ has a natural structure of a symplectic manifold.
A Lagrangian connection $D$ on $M$ is a $n$ dimensional distribution for $TM$ such ...
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$L^2$-Euler number
Suppose $M$ is a closed manifold, and $\tilde M$ is the universal covering.
Q: Can we say that $\chi(M)=L^2\chi(\tilde M)$, where $L^2\chi(\tilde M)$ denotes the alternative sum of the dimension of $...
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Distance function on a curve on a manifold
Suppose that we are given a non-negative even function $b\in C^\infty[-1,1]$ satisfying $b(0)=0$, $\sqrt{b(x+y)}\le \sqrt{b(x)}+\sqrt{b(y)}$ for any $x,y\in[-\frac12,\frac12]$. Can we always find a 3-...
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Volume growth of balls implies volume growth of spheres?
Suppose I have a complete, non-compact Riemannian manifold $M$ such that the volume of balls around a fixed point $p \in M$ satisfies
$$\mathrm{vol}(B_R(p)) \leq v(R)$$
for some function $v$. Can we ...
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950
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Continuation (extension) of harmonic functions
Suppose $(M,g)$ is a simply connected smooth Riemannian manifold with smooth boundary and suppose that $U \subset \partial M$ is a smooth open connected subset of the boundary. Now my question is the ...
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The first eigenfunction of Dirac operator for surface
Let $M$ be a spherical oriented surface with Riemannian metric and with trivial spin structure. We know that the equation $$D \phi = \rho \phi$$, where $\rho: M\rightarrow \mathbb{R}$ is a real scalar ...
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2
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2
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Invertible (isometric) sections of certain hom bundles over sphere
Assume that we have a vector bundle $E$ over $S^n$.
Is there a continuous family of invertible linear maps $T_x:E_x \to E_{-x}$?
Here continuity has the obvious meaning as soon as ...
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About the ergodic theorem of Birkhoff in the context of a compact Riemannian manifold
Let $V$ be a compact Riemannian manifold, $G$ the set of diffeomophisms of $V$, let $\nu$ be a probability measure in $G$. Suppose that $\exp_{x}$ is diffeomorphism in $\mathcal{B}_{2I}(x)\subset V$ ...
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A quantity associated with a Riemannian surface
Assume that $E$ is a Riemannian vector bundle, then its structure group is reduced to $O(n)$. Then the structure group of $E \oplus E$ is reduced to $D(O(n) \oplus O(n)) \subset Sp(2n)$ ...
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Is $\delta(df \wedge df)=0$ an Euler-Lagrange equation?
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Primary definition of a geodesic
I am wondering if there is a sense in which one of these definitions
for a geodesic on a smooth Riemannian manifold is primary to the other.
A geodesic has acceleration zero, i.e., it is self-...
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2
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1
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Relation between the geodesics of Finsler norms $F(V)$ and $F(-V)$
I am trying to solve this exercise. Let $(M,F)$ be a Finsler space and define $\tilde{F}(x,y):=F(x,-y)$. Then $(M,\tilde{F})$ is a Finsler space and given a geodesic $t\mapsto \gamma(t)$ of $F$, $t\...
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''Are Hermitian metric pullbacks automatically via biholomorphisms?''
The awkward title is an attempt at approximating the following specific question: Let $(M^{2n}, J)$ be a complex manifold, suppose $g_0$ is a Riemannian metric $M$ compatible with $J$, and suppose $\...
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Existence of geometric tubular neighborhoods in Finsler spaces
$\DeclareMathOperator\Tub{Tub}$I have not found any reference among the well-known books about the existence of a geometric tubular neighborhood in the Finsler spaces. I am wondering if there exists ...
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Smoothness of some power of the geodesic distance in a Finsler geometry
I know that generally the geodesic distance $d_x$ from a point $x$ in a Finsler space is not smooth ($C^\infty$). According to Shen, the square of it is just $C^1$ at $x$. Now I am wondering if there ...
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Calibrated submanifolds in Spin(7) and Calabi-Yau threefold
Suppose I have a Cayley cycle in a $Spin(7)$ holonomy manifold $M$, i.e. a calibrated submanifold. In the special case that $M=CY_3\times T^2$, is it possible that the Cayley cycle reduces to a Kahler ...
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1
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Vector field with Harmonic flow
Assume that $(M,g)$ is a Riemannian manifold. A vector field $X$ on $M$ is called a harmonic vector field if the corresponding $1$-form $\alpha$ with $\alpha(Y)= \langle X,Y \rangle_g$...
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Geometric and holomorphic structure of $\mathbb{C} \rtimes \mathbb{C} \setminus \{ 0 \}$
Put $G= \mathbb{C} \rtimes_{\phi} \mathbb{C} \setminus \{0\}$ where $\phi_{a} (z)= az$ for $a \in \mathbb{C} \setminus \{0\}$.
$G$ is a real $4$ dimensional Lie group; then it has a unique left ...
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2
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General questions on the eigenfunctions of Laplacian and Dirac operators
We know that the eigenvalues of the Laplacian contains a lot of information of a Riemannian manifold, but they do not determine the full information ( Hearing the shape of a drum). And the ...
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About pairwise distances of some points in a Riemannian manifold $M$ of ${\rm sec}\ M\geq 1$
This question is cross-posted in MO and MSE https://math.stackexchange.com/questions/2276064/about-pairwise-distances-of-some-points-in-a-riemannian-manifold-m-of-rm-se
Assume that there are points ...
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Isoperimetric inequality in CAT(0) surfaces
I am looking for a version of the (2-dimensional) isoperimetric inequality for globally CAT(0) (in particular simply-connected) surfaces. I am particularly interested in characterizing the disk of ...
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A non vanishing vector field compatible to a Riemannian metric
Assume that $(M, g)$ is a connected Riemannian manifold which is either open or is compact with zero Euler characteristic.
Is there a non vanishing vector field $X$ on $M$ such ...
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2
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Are all symmetries of the Dirichlet functional isometries?
This is a cross-post from MSE (no answer there).
Let $M,N$ be oriented $d$-dimensional Riemannian manifolds, $M$ compact*, and let $f:M \to N$ be smooth.
Consider the Dirichlet energy functional: $...
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A characterization of flat metrics via global vector fields
Let $(M,g)$ be a Riemannian manifold with $LC$ conncection $\nabla$.
Assume that for every three global vector fields $X,Y,Z \in \chi^{\infty}(M)$ with $[X,Y]=0$ we have $\nabla_{X} \nabla_{...
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Riemannian Manifolds of Bounded Curvature
I am a complete newbie Riemannian Geometry with a particular application in mind so please excuse a lack of rigor in the question.
Suppose I have a manifold with sectional curvature everywhere ...
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De Turck trick on mean curvature flow
I am reading the book Lecture on mean curvature flow by Xi-Ping Zhu.
Suppose $M^n$ is an n-dimension smooth manifold and $X(x,t):M^n \rightarrow R^{n+1}$ be a one-parameter family of smooth immersion....
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Negatively curved manifolds with many totally geodesic submanifolds
I'm curious about the following question and have not been able to find any literature on the topic: Suppose that $M$ is a closed negatively curved Riemannian manifold with a "large" quantity of ...
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Is this expression for the Laplacian of conformal maps between Riemannian manifolds known?
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Convex hull of a connected subset on a complete surface of non-positive curvature
Let $S$ be a simply connected surface, possibly with boundary components, with a smooth complete metric of non-positive curvature. Let $X\subset S$ be a closed connected subset. I would like to know ...
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Polar coordinates of a set with different radius and angle
Let $M$ be a $2$-dimensional Riemannian manifold and let $U\subset M$ be an open set. Suppose there exist polar coordinates $(r,\theta)$ with center $q\in M$ such that
$$U=\lbrace{ (r,\theta): 0<...
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Sources for Alexandrov surfaces
There are two distinct notions in differential geometry associated
with A. D. Alexandrov: (1) Alexandrov spaces of courvature bounded
from below; (2) Alexandrov surfaces of bounded total curvature (...
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2
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Conformal harmonic maps in high dimensions are scaled isometries
This is a cross-post from MSE (where I got no answer).
It is well-known that conformal maps between $2$-dimensional Riemannian manifolds are harmonic.
I discovered lately that in dimension $d>2$, ...
- 6,621
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0
answers
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Exponential map for non-smooth Finsler manifolds
Context
I'm interested in studying reversible Finsler manifolds which do not have the strong convexity of the Hessian property (that is the Finsler function is a regular norm on every tangent space). ...
- 4,969
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0
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"Riemannian" collar theorem
Let $(M,g)$ be a compact manifold of dimension $n$ with boundary. If $\partial M$ is smooth then one has a control on the determinant of the Jacobian of the diffeomorphism in the collar theorem, i.e....
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3
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Automatic transfer of pointwise metric computations to bundle computations
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Smoothness of the square of the distance function on a Riemannian manifold
Let $(M^n,g)$ be a smooth Riemannian manifold. The distance between two points is the infimum of the lengths of the curves which join the points. Consider the square of the distance function
$d^2\...
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Geodesics equation on Lie groups with left invariant metrics
First of all, I am so sorry if this question is not appropriate to be here. I tried to ask something similar on Math Stack Exchange but it didn't have much attention. Any comment and I delete the ...
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2
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Tube formula for a hypersurface in a Riemannian manifold
Let $(M,g)$ be a complete Riemannian manifold and $N$ a closed (orientable) hypersuface of $M$. Let $d$ be the signed distance from $N$ and $N_r=\{x\in M: 0<d(x,N)<r\}$. For $r$ small enough $$\...
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SO(3) action on (simply connected) 6 manifold with discrete fixed point
If a 6-dimensional orientable smooth manifold $M$ admits a smooth effective $SO(3)$ action with discrete fixed point set, what can we say about the topology of $M$? What if we assume that in addition ...
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What does positivity of the first Pontryagin number of a vector bundle tell us?
Some context:
In the theory of compact, oriented Riemannian Einstein 4-manifolds, there is a a fundamental topological constraint that is implied by the Einstein equations. To wit, if $\chi$ and $\...
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A step in the proof on the uniqueness of mass
I am reading the survey paper "The Yamebe Problem" by Lee and Parker. In section 9, Theorem 9.6 in P.78, it was proved that the mass is well defined in the sense that $m(g)$ depends only on the metric ...
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3
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Parallel transport as algebra isomorphism
Assume that there is an smooth structure of the matrix algebra $M_{n}(\mathbb{R})$ on fibers of the tangent bundle of a $n^2$ dimensional manifold.
Is there a Riemannian metric on $M$ such that all ...
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votes
1
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Isomorphism of Complex Stiefel manifold and Homogeneous space of unitary group, and the Stiefel logarithm problem
It is well known that $U(n)/U(n-k) \cong V_k(\mathbb{C}^n)$ where $U(n)$ is the unitary group, and $V_k(\mathbb{C}^n)$ is the appropriate Stielfel manifold.
I further understand that $V_k(\mathbb{C}^...
- 2,069
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4
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Why do people study curve shortening flows?
I've recently been studying Riemannian geometry with goal of studying and doing research in Ricci flow, however, I've been noticing that a lot of work in Riemannian geometry seems to be done in ...
- 1,207
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1
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Reverse Toponogov triangle comparison
See the wiki page https://en.wikipedia.org/wiki/Toponogov%27s_theorem
One consequence of the Toponogov comparison Theorem is that if the sectional curvature of a manifold $M$ is pinched below by a ...
- 319
0
votes
2
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503
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A relation between gradient vector field and Hamiltonian vector field
Assume that $U$ is an open subset of $\mathbb{C}^n$. We fix the standard almost complex structure $J$ on $U$.
Assume that $\omega$ is an arbitrary symplectic structure on $U$.
Is there a Riemannian ...
- 312
3
votes
1
answer
150
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Foliation by Asymptotic lines
Suppose $(M,g)$ is a compact Riemannian manifold with boundary. I am interested about existence of surfaces $\Gamma$ embedded in $M$ with the following property:
$\Gamma$ is foliated by geodesics (...
- 4,089
0
votes
2
answers
499
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Lie algebra of Gradient vector fields(2)
Motivated by this question, is there a $n$ dimensional Riemannian manfold $M$, $n>1$ such that the space of all gradient vector fields is a Lie algebra under the usual Lie bracket ...
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