New answers tagged dg.differential-geometry
4
votes
Normalizer of solvable linear group is an algebraic group?
The normalizer of every connected closed subgroup of $\mathrm{GL}_n(\mathbf{R})$ is the stabilizer of its Lie algebra, so is Zariski-closed.
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What does it mean that the Hessian is proportional to the metric?
Since the proof that such a function implies the metric is a warped product metric is fairly simple, I include a complete copy below.
1
Start with $\nabla^2 f = \lambda g$. For any vector field it ...
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2
votes
Accepted
Does any warped product metric admit a function with hessian proportional to the metric?
Yes. (With the caveat that if there is a function such that $\nabla^2\varphi = \psi g$, then necessarily it is a warped product over a one-dimensional base, so your question should really require $\...
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2
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Relationship between the holonomy pseudogroup and holonomy homomorphism (foliation)
I will answer even though this question was asked a year ago. Hopefully my answer is of some help.
Yeah that's right.
I don't quite understand the question, the Remark 2.7 says that any transverse ...
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1
vote
Lower bound for domain of exponential map on Lorentzian manifolds
You've waited a long time for an answer. And I am surprised that no one has written one.
Let $M=\mathbb{R}\times(0,\infty)$. The exponential map isn't defined at $(x,t)$ for vectors $(u,s)$ with $s<...
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4
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Compactification of a product of manifolds
Yes, the quotient $C_M = \overline{M} \times \overline{\mathbb{R}} / \left\{\{x\} \times \overline{\mathbb{R}} : x \in \partial\overline{M}\right\}$ seems to do the job, where I mean that the points ...
- 19.7k
2
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Lie group framing and framed bordism
Lie group framing is a reference to the group action. A Lie group $G$ acts on the left of $G$ by the map
$$(g,h) \longmapsto gh.$$
Similarly there are actions on the right, and conjugation actions, ...
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8
votes
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Bounded covariant derivative of curvature tensor
I did not find any reference the proves a theorem as strong as stated in the question.
Greene-Wu and Stefan Peters indepdently proved the following using almost linear coordinates (defined and studied ...
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6
votes
Convergence of metric spaces of increasing dimension
Theorem. For all integers $0<m<n$ $d_{GH}(\mathbb{S}^m,\mathbb{S}^n)\geq \frac{\pi}{4}$.
This is Theorem B on page 7 in
S. Lim, F. Mémoli, Z. Smith, The Gromov-Hausdorff distance between ...
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2
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How do we calculate the gradient of this function defined using the Riemannian logarithm on a Riemannian manifold?
In order to differentiate something containing the logarithm, it's best if you can differentiate the logarithm itself.
And because you work in a neighborhood where the exponential map is a ...
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3
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Accepted
Convex hull of 3 points in Cartan-Hadamard manifolds
I believe that the idea described by Ian Agol works, and can be elaborated on as follows. The general fact we want to establish is that the convex hull of a finite collection $X$ of points in $M$ is ...
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7
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string bordism group and framed bordism group for $d \leq 6$ and $d \geq 7$
The map $i : * = B\{e\} \to BString$ over $BF = BGL_1(S)$ is $7$-connected, so induces a $7$-connected map $S = M\{e\} \to MString$ of Thom spectra, by the Thom isomorphism and Hurewicz theorem. At ...
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Convergence of metric spaces of increasing dimension
Gromov has such a general result in
Gromov, Misha. Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. ...
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5
votes
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Derive distributional inequalities from pointwise estimates
First note that the result is true if the support of $\varphi$ is disjoint from $E$; this follows from integration by parts.
Next note that every algebraic set is a finite union of smooth submanifolds ...
- 35.2k
2
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Manifolds with nonpositive radial curvature
Assume that $M$ is three-dimenional and $\exp_p\colon\mathbb{R}^3\to M$ is a diffeomorphism.
Let $g_r$ be the Riemannian metric induced on $\mathbb{S}^{2}$ by the map $x\mapsto \exp_p(r\cdot x)$.
Note ...
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2
votes
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Decompositions of $\partial_i$ to the radial direction and rotations in higher dimensions
The formula you gave in the beginning is a special case of the vector triple product formula in $\mathbb{R}^3$
$$ a\times (b\times c) = (a\cdot c) b - (a\cdot b) c $$
Let $v$ be an arbitrary vector ...
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4
votes
Accepted
Examples of hyperbolic manifolds of dimension $\geq$ 3 with disjoint totally geodesic hypersurfaces
I do not have a self-contained reference, but the key is
Long, D. D.; Reid, A. W., Constructing hyperbolic manifolds which bound geometrically, Math. Res. Lett. 8, No. 4, 443-455 (2001). ZBL0992.57023....
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2
votes
Accepted
Is any globally-hyperbolic manifold conformally equivalent to one with complete slices?
This is the original Nomizu-Ozeki article:
Nomizu, Katsumi; Ozeki, Hideki, The existence of complete Riemannian metrics, Proc. Am. Math. Soc. 12, 889-891 (1961). ZBL0102.16401.
Applying their proof ...
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1
vote
Accepted
Curve length in the Sasaki metric
For the inequality: first note that if $Y$ is a vector field parallel along $\alpha$, then
$$ \tau_\alpha (X + Y)(\alpha(0)) - (X+Y)(\alpha(1)) = \tau_\alpha X(\alpha(0)) - X(\alpha(1)) $$
and
$$ \...
- 35.2k
10
votes
How to tackle the smooth Poincaré conjecture
Fastforward $11+\frac12$ years, I thought I'd mention that I tried to do what my original post suggested, which is when I started my PhD and then when I completed my thesis it naturally spawned this ...
- 16.9k
2
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Developable 3-manifolds in $\mathbb{R}^4$
In this paper, I showed that an open and dense subset of any flat ruled submanifold (and, in particular, of any flat hypersurface without planar points) is a union of cylindrical, conical, and tangent ...
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3
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reference for reading Schoen Yau positive mass theorem proof II
Already found one that might be helpful so I share it here, notes for a short course by Schoen at Tsinghua unversity, https://web.math.ucsb.edu/~zhou/Math_GR_2012.pdf
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0
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Torsion-free Cartan connections
In my Introduction to Cartan geometries, p.30, I define a near parallelogram, a 5-sided figure, of constant vector fields flows, say of two elements $A,B$ in the Lie algebra of the model, to be a ...
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1
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Calculation of the top Chern class of spinor bundle over $S^{2n}$
I will present an explicit calculation using Chern-Weil theory, which makes an amusing use of Legendre's duplication formula for the gamma function.
The Chern character form of a vector bundle $E$ ...
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2
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Manifolds with negative dimension – Definition, References
One place where negative dimensional manifolds appear naturally is complex cobordism $U^*$. Intuitively, elements of the abelian group $U^n(X)$ are represented by families of $(-n)$-dimensional ...
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4
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Does Hermite-Einstein imply Kähler-Einstein?
If one interprets the OP's question literally, the answer is 'yes', but I imagine that the OP didn't literally mean what the OP wrote.
First, interpret everything literally: Assume that $(M,g,\omega)$...
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2
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Going from piecewise to genuine geodesic without decreasing number of intersections?
Here's a counterexample.
Take a cone of unrolled angle strictly between $\frac\pi2$ and $\pi$. Smooth the vertex and take $\gamma$ to be a radial ray far from the vertex. Then you can make a piecewise ...
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Modern mathematical books on general relativity
R.K. Sachs and H. Wu, General Relativity for
Mathematicians
is a valuable source.
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