New answers tagged riemannian-geometry
2
votes
$2\mathrm{d}$ area minimizing short embeddings
If think yes, because the assumption $f^*h \leq g$ gives pointwise bounds on the derivatives of $f$ : in local charts, $|\partial_i f|^2 \leq g_{ii}$.
So up to translation, you can put the image of $f$...
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10
votes
Accepted
Can we prescribe the $L^2$ norm of the scalar curvature on a four-manifold?
This is not always possible.
Let $M$ be a compact smooth manifold of dimension $n$. Consider the Einstein-Hilbert functional $\mathcal{E}$ given by
$$\mathcal{E}(g) = \dfrac{\displaystyle\int_Ms_g d\...
- 17k
12
votes
The convex hull of a manifold whose cobordism class is trivial
Implicit in the other responses is the fact that if $M$ bounds a convex manifold $W$, then $W$ is contractible and so M has the homology of a sphere. So any null-cobordant manifold that is not a ...
- 17.1k
8
votes
The convex hull of a manifold whose cobordism class is trivial
There are exotic spheres (which are null cobordant) which do not bound a parallelisable manifold. Since the convex hull is contractible, it would be parallelisable if it were a manifold, so these guys ...
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2
votes
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Closed almost geodesics in a Riemannian manifold
Any curve $\gamma:[a,b]\to M$ parametrized by arc length is an $\varepsilon$-geodesic for any $\varepsilon>0$.
The inequality $(1-\varepsilon)dist_M(\gamma(x),\gamma(y))\leq length[\gamma(x),\gamma(...
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1
vote
Explicit computation of the vertical and horizontal vector bundles
I am just answering to (1) and (2): The horizontal bundle is defined as follows. Consider a point $v\in T_pX\subset TX,$ and $w\in T_pm$. Consider a curve $\gamma$ through $p$ with $\gamma'(0)=w.$ ...
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1
vote
Accepted
Cross product of two infinitesimal bendings
Let $M$ be a rotationally symmetric cylinder, $Z_1$ be a vector field tangent to M generating the isometric translation along the cylinder and $Z_2$ be a vector field also tangent to M generating the ...
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1
vote
What are good Morse Theory lecture notes and books?
also in the M. W. Hirsch, Differential topology ,Chapter 6 : Morse Theory.
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10
votes
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Does the first Laplacian eigenfunction on a homogeneous space have a unique maximum?
The flat torus $\mathbb{T} = \mathbb{R}^2/\Lambda$ gives a counterexample: The first nontrivial eigenvalue is of the form $\lambda_1 = \xi_1^2+\xi_2^2$, where $\xi = (\xi_1,\xi_2)$ is a nonzero ...
- 97.6k
1
vote
Accepted
Convergence of extremal subsets in Alexandrov spaces
The limit of extremal subsets is an extremal subset, see Lemma 4.1.3 in Petrunin's Semiconcave functions in Alexandrov geometry. The non-collapsing assumption is not needed.
- 26.6k
0
votes
Area of a deformation of a closed surface
The answer to the second question is still negative.
Let $M^3$ be the cylinder $\mathbb{S}^2 \times \mathbb{R}$. Then $\Sigma = \mathbb{S}^2\times \{0\}$ is totally geodesic in $M$, and so for any ...
- 30.7k
2
votes
Accepted
A problem arising from reading a lecture on the Yamabe problem of how the Hölder inequality is used
Application of Holder's inequality
Notice that the estimate on the Green's function means that
$$ \int |G(P,Q)|^\alpha ~dQ $$
is bounded whenever $\alpha < \frac{n}{n-2}$ (and the bound can be ...
- 30.7k
1
vote
Accepted
Weitzenböck formula and comparison of norms
I am not an expert in all the delicate points of clifford algebras and spin structures, but I think the following shows the constant can't be 1 in general: let $(M,g)$ be a Riemannian manifold of ...
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5
votes
Harmonic functions on complete Riemannian manifolds
For the $n$-dimensional hyperbolic space it is already the space of bounded harmonic functions that is infinitely dimensional, which follows from the integral Poisson formula.
- 15.4k
1
vote
Decomposition of tensors
If what @AMath91 meant by a codazzi tensor is a symmetric tensor $\lambda \in \Gamma(S^{2}T^{*}M)$ satisfying $d^{\nabla}\lambda=0$ then I think the decompostion in the question cannot be true. Take ...
- 336
5
votes
Accepted
Why are we interested in spectral gaps for Laplacian operators
A spectral gap gives information on geometry of the manifold via Cheeger's inequality, https://en.wikipedia.org/wiki/Cheeger_constant See also Buser's inequality discussed there. More directly, a ...
- 11.3k
2
votes
Is it known whether a closed simply-connected manifold of non-negative curvature admits positive Ricci?
No. Every simply connected compact manifold of nonnegative sectional curvature admits positive Ricci curvature. This has been proven by Boehm--Wilking: https://link.springer.com/article/10.1007/s00039-...
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