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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.
18
votes
1
answer
908
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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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16
votes
2
answers
2k
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Is the Gromov conjecture still open?
Today I read about Gromov's definition of minimal volume for smooth manifolds.
$$\min {\rm Vol}(M):=\inf_{|K_g|\leq1}\{{\rm Vol}(M,g)\}.$$
Gromov's conjecture states that for every closed simply con …
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14
votes
2
answers
1k
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What are the important geometric-topological consequences of 4-dimensional version of Gauss-...
The Gauss–Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry to their topology and has very important applications to Riemann surface theory …
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14
votes
Accepted
Information about Milnor conjecture
According to David Roberts comment and the following paper it is open for dimensions $n\geq 4$.
Pan, Jiayin, A proof of Milnor conjecture in dimension 3, J. Reine Angew. Math. 758, 253-260 (2020). ZBL …
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11
votes
1
answer
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Geometric interpretation of horizontal and vertical lift of vector field
In many References such as D.E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds chapter 9, and Differential Geometric Structures
By Walter A. Poor Page 54; the horizontal and vertical l …
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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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9
votes
1
answer
406
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Almost Complex manifolds of constant curvature
Edited (after R. Bryant comment)
Let $(M,\cal J,g)$ be a almost Hermitian manifold (not necessary integrable). i.e., ${\cal J}^2=-I$ and $g({\cal J} X,{\cal J} Y)=g(X,Y)$. Suppose that $\{X_i,{\cal J …
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8
votes
0
answers
405
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What specifically is the gap in Aubin's argument about positive Ricci curvature that Paul Eh...
In his paper [2], Paul Ehrlich write
In [1], Aubin stated a theorem which implied as a corollary that if a manifold
$M$ admits a Riemannian metric with nonnegative Ricci curvature and
all Ricci curva …
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6
votes
1
answer
227
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Does $\pi_k(M)\neq 0$ implies $\operatorname{ind}(\gamma) < k$?
Cross post from MSE. and sorry if this is an obvious question.
Here is a line of proof of Theorem 1.15 from
Brendle, Simon, Ricci flow and the sphere theorem, Graduate Studies in Mathematics 111. Prov …
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6
votes
1
answer
252
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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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6
votes
1
answer
2k
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Relation between harmonic vector field and harmonic 1-form
Definition 1: A unit vector field $X$ side to be harmonic if it is critical point for the following energy function
$$E(X)=\frac{1}{2}\int_M\|dX\|^2dvol_g=\frac{m}{2}vol(M,g)+\int_M\|\nabla X\|^2 …
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4
votes
1
answer
208
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Existence non-trivial parallel $p$-form implies non-triviality of $p$-th cohomology group us...
Cross-post from MSE.
Suppose $(M,g)$ be a closed Riemannian manifold. Because every parallel (nontrivial) $p$-form $\omega$ is harmonic so the $p$-th Betti number should be positive i.e. $b_p\geq 1$. …
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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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4
votes
1
answer
309
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Existence parallel vector fields and its effect on the topology of manifolds (Karp's Thesis)
It seems that there is no digital copy of Leon Karp's Ph.D. thesis
L. Karp, Vector fields on manifolds, Thesis, New York Univ., 1976.
on internet and his paper excerpted from his thesis is very brief …
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4
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2
answers
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When a Killing vector field on Riemannian manifold $(M,g)$ is gradient?
Let $(M^n,g)$ be a Riemannian manifold that admit a unit Killing vector field $X$. i.e., $\mathscr{L}_Xg=0$. Is it possible that there exist a smooth function $f$ on $M$ such that $X=\mathrm{grad}f$?
…
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