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Results tagged with riemannian-geometry
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user 36688
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.
23
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1
answer
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Eigenvalues of Laplace operator
Assume that $(M,g)$ is a Riemannian manifold.
Is there any relation between the sequence of eigenvalues of Laplace operator acting on the space of smooth functions and the sequence of eigenvalues of …
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18
votes
3
answers
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Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order deriv...
EDIT: According to some comments on this post I revise the title to remove the misunderestanding.
Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated t …
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17
votes
2
answers
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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 inv …
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17
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Finding a 1-form adapted to a smooth flow
If the vector field is geodesible then such $1$ - form exists.
A geodesible flow on a manifold $M$ is a one dimensional foliation associated with a non vanishing vector field such that …
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13
votes
5
answers
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A geometric proof of the Gauss-Lucas theorem
Motivated by a geometric proof of the Fundamental Theorem of Algebra we ask:
Is there a geometric proof for the Gauss-Lucas theorem? Since we are working on a half plane, can one imagine a possible p …
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12
votes
3
answers
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Limit cycles as closed geodesics (in negatively or positively curved space)
Updated 1/25/2023 I just added a related post below:
Jacobi fields, Conjugate points and limit cycle theory
EDIT: Here is a related post which concern quadratic vector fields rather than Van de …
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10
votes
3
answers
713
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Number of disjoint simple closed geodesics
According to Jairo comment on the first version of this question I revise the question as follows;
Let $g$ be a real analytic Riemannan metric on $S^{2}$. Is it true to say that:
There are at most a …
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10
votes
1
answer
308
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A characterization of round sphere
Let $(M,g)$ be a $k$ dimensional compact Riemannan manifold which is isometrically embeded in $\mathbb{R}^{k+1}$. The distance arising from the Riemannian metric is denoted by $d_g$.The Euclidian dist …
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9
votes
1
answer
676
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Does every manifold admit a Lagrangian Riemannian metric?
Let $(M,g)$ be a Riemannian manifold. The $LC$ connection associated to the metric gives an $n$ dimensional distribution $D$ for $TM$. Let $\omega$ be the symplectic structure of $TM$ which is obtain …
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8
votes
3
answers
411
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Extension of a vector field to an orthonormal frame for a flat metric
Assume that $U$ is an open set in the plane and $X$ is a non vanishing vector field on $U$.
Is there a non vanishing vector field $Y$ on $U$ such that the pair $\{X,Y \}$ plays the role of an orthono …
- 356
8
votes
1
answer
364
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Can a harmonic vector field possess a limit cycle?
Can a harmonic vector field $X$ on a Riemannian surface $(M,g)$ possess a limit cycle(An isolated periodic orbit)?
Note that the Laplacian of a vector field is defined via natural correspondence bet …
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7
votes
2
answers
357
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Is every Lie subgroup of a Lie group isometric to all its conjugates?
Let $G$ be a Lie group with a left invariant metric. Assume that $N$ is a Lie subgroup of $G$.
For a given $g\in G$, are $N$ and $g^{-1} N g$ necessarily isometric Riemannian manifold when they inher …
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7
votes
1
answer
335
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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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7
votes
0
answers
519
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Limit cycles as closed geodesics(2)
Hilbert 16th problem asks for a uniform upper bound $H(n)$ for the number of limit cycles of a polynomial vector field of degree $n$ on the plane. Here is an updated proof of the fi …
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7
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1
answer
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Hilbert 16th problem via hyperbolic geometry
More than 16 years ago, I heard from someone that he thinks that there is a possible relation between Hilbert's 16th problem(for $n=2$) and Hyperbolic geometry. He says that a po …
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