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Results tagged with riemannian-geometry
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user 90655
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.
1
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Positive sectional curvature does not imply positive definite curvature operator?
This argument is due to Burkhard Wilking. The other answers, answer to the OP question very well but I want to give a sharp example. I.e. a manifold with positive sectional curvature and curvature op …
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1
vote
1
answer
184
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Why can we construct the following method?
I read in some papers the following method:
Let $h$ be a $(1,1)$-tensor field on 3-dimensional Riemannian manifold $(M,g)$ and $p\in M$. Then there exists a smooth local orthonormal basis of the for …
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2
votes
1
answer
686
views
Manifolds with positive sectional curvature
In the study of manifolds with positive sectional curvature, I guess the following statement is true:
Let $M$ be a manifold with positive (but not necessarily constant) sectional curvature. Then $ …
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6
votes
1
answer
252
views
A metric has positive sectional curvature if and only if ${\rm Ric}_{ij} < \frac{r}{2}g_{ij}$
This is a cross-post from my question on MSE.
It is well known that
In dimension three a metric has positive sectional curvature if
and only if ${\rm Ric}_{ij} < \frac{r}{2}g_{ij}$. where $r$ …
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18
votes
1
answer
908
views
Consequences of Gromov's Conjecture
In Peter Petersen words, Gromov Betti number estimate is considered one of the deepest and most beautiful results in Riemannian geometry; which asserts that
Theorem (Gromov 1981). There is a constant …
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2
votes
1
answer
170
views
An equivalent definition for contact pair manifolds
َContact manifold
A $(2n + 1)$-dimensional manifold $M$ is said to be a contact manifold if it
admits a global 1-form $\eta$ such that $\eta\wedge (d\eta)^n\neq 0$.
There is an equivalent defi …
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1
vote
1
answer
546
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Under what condition the covariant derivative of Ricci operator along Killing vector field v...
Let $(M,g)$ be a Riemannian manifold and $V$ a unit Killing vector field on it. Under what condition on curvature tensor the following equation hold:
$$\nabla_VQ=0,$$
where $Q$ is the Ricci operator d …
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1
vote
0
answers
156
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Proof of Berger in "Sur les variétés d’Einstein compactes"
I would appreciate any reference that contains either a translation or proof of the following interesting observation of Berger (Sur les variétés d’Einstein compactes, M. Berger paper (in French)).
…
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4
votes
What are the important geometric-topological consequences of 4-dimensional version of Gauss-...
A good paper in this direction is "Some implications of the generalized Gauss-Bonnet theorem" written by Bishop and Goldberg which proved the following two theorems
Theorem 1.1. A compact and o …
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-3
votes
1
answer
374
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Is the 2-dimensional Gauss-Bonnet theorem applicable in higher dimensions? [closed]
This is a cross-post of this MSE post that users commented that it is appropriate for MO.
I want to know
Question: Is the 2-dimensional Gauss-Bonnet theorem applicable (any topological or geometrica …
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2
votes
1
answer
270
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Is there any Riemannian manifold of zero dimensional isometry group such that
Sorry if this question is belongs to MSE. I have no idea about it.
Question: Is there any Riemannian manifold of zero dimensional isometry group which its Ricci curvature is positive (or maybe zer …
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9
votes
Deforming metrics from non-negative to positive Ricci curvature
This is not a complete answer but would be helpful. Here are a few facts:
Theorem (T. Aubin 1970 and P. Ehrlich 1976). If the Ricci curvature of a compact Riemannian manifold is
non-negative and posit …
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1
vote
1
answer
323
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Conformal Killing vector field on contact manifolds
An interesting class of contact manifolds is the class of $K$-contact manifolds ($\mathcal{L}_\xi g=0$) which have been studied by many authors. It is natural to study conformal Killing-contact manifo …
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3
votes
0
answers
238
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About Riemann curvature tensor of local reflection
Let $\alpha: [a,b]\to M$ be an embedded curve in a Riemannian manifold $(M,g)$ and let
$p$ be a point in $M$, not on the curve $\alpha$. If $p$ is close enough to $\alpha$, there exists a
unique geode …
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3
votes
0
answers
342
views
Euler-Lagrange equations for $p$-Harmonic vector fields
Harmonic vector fields are critical points of Dirichlet energy function on the set of all unit vector fields on $M$, which is defined as follows:
$$E(X):=\frac{1}{2}\int_M\|dX\|^2\mathrm{dVol_g}\qquad …
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