Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
0
votes
0answers
41 views
Riemannian metric on complexification of Lie group
Let $G$ be a compact linear group and $G^c$ be its complexification. Then there is a diffeomorphism $f: G^c \to G \times Lie(G) $ given by $$ x e^{iA} \to (x,A).$$
Let $h$ be the pull back metric of ...
0
votes
1answer
34 views
Decomposing connections on extensions of the frame bundle
I have posted this question on math.stackexchange, without success. I'll make it brief:
Let $E\rightarrow M$ be an orientable vector bundle of rank n equipped with some Riemannian metric, ...
5
votes
2answers
148 views
A Scalar Curvature Computation in Brendle Marques Neves' Min-Oo Conjecture paper
I'm reading a paper on the Min-Oo Conjecture (http://arxiv.org/abs/1004.3088), and I'm stuck on the following step in a proposition:
Given a metric $g_0(t)$ on the upper hemisphere $\mathbb{S}^n_+$, ...
0
votes
0answers
53 views
How to define the distributional Hessian for a convex function on a $C^0$ Riemannian manifold?
M is a $C^1$ manifold with a $C^0$ Riemannian metric, f is a convex function on M. How to define a functional on M which can represent $Hessf$?
For example: for $\Delta f$ we can define the ...
-2
votes
1answer
96 views
Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field? [on hold]
Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...
-1
votes
0answers
79 views
Time derivative of an integral on a moving surface? [on hold]
I need to take the time derivative inside the surface integral,
$$\displaystyle\dfrac{\mathrm{d}}{\mathrm{d}t}\left(\oint_{\partial B} \left(\mathbf{x} \times ( \mathbf{n} \times \mathbf{u} ) ...
3
votes
0answers
125 views
smoothing a current
Let $M$ be a smooth oriented manifold of dimension $n$ and $T$ a current of dimension $k$ on $M$. Let $\phi:P\times M \to M$ be a proper smooth family of diffeomorphisms of $M$ (i.e. $P$ is a smooth ...
0
votes
2answers
106 views
lift of Riemannian metric to branched double cover
Let $\hat{M}$ be a branched double cover of $M$. Is there a way to lift a Riemannian metric $g$ on $M$ to get a smooth Riemannian metric $\hat{g}$ on $\hat{M}$. Moreover, if $g$ has nonnegative ...
-4
votes
0answers
74 views
how to make Contravariant and Covariant tensors applicable to problems of curvatures in halfspace problems? [on hold]
Consider a material halfspace and assume it to be made of infinite number of layers of same material, such that when the material is loaded at the top surface, how to quantify the variation of ...
0
votes
1answer
113 views
extension of Riemannian metric on real affine variety
Given a Riemannian metric $g$ on the real part $X_R$ of a real affine variety $X$,
is there a "natural" way to extend $g$ to be a Riemannian metric on $X$?
-1
votes
0answers
62 views
how to construct 3D curve in highway geometric design [on hold]
give you some control points ,also give you the initial point and final point ,their curvature ,torsion and coordinate are kown.How to construct a three-dimensional space curve under the constaint of ...
5
votes
1answer
145 views
Question about conjugate points
If there exist two geodesics from $p$ to $q$ that are not only different from each other but also infinitesimally close to each other, then it implies that $q$ is conjugate to $p$.
Can anyone give an ...
2
votes
1answer
156 views
If there exists a nontrivial vector field $V$ such that $\nabla_{X}V=0$ for any vector field $X$, the manifold must be flat?
If there exists a nontrivial vector field $V\not=0$ in Riemannian manifold $M$ and an open set $U\subset M$ such that $\nabla_{X}V=0$ in $U$ for any vector field $X$ in $M$, then dose $U$ have to be ...
0
votes
0answers
39 views
Factor of 2 In the Definition of Metric Contact Structure
In Blair's book and many many literatures, I see definition of a contact metric manifold which involves a relation
\begin{equation}
d\kappa \left( {X,Y} \right) = g\left( {X,\Phi Y} \right)
...
4
votes
2answers
167 views
When distance nonincreasing map is an isometry
Let $f: M \to M$ be a distance nonincreasing map between a closed Riemannian manifold $M$
and $f$ is homotopic to the idendity map. Is it then $f$ an isometry?
4
votes
1answer
177 views
Quotienting $SU(3)$ by $U(1)$?
As is well-known, if we quotient $SU(2)$ by the action of $U_1$, embedded in the diagonal as $(e^{i \theta}, e^{-i \theta})$, we get the $2$-sphere. As is also well-known, if we quotient $SU(3)$ on ...
0
votes
0answers
90 views
Jacobian change of basis matrix for different dimensions
I am considering a real Lie group $G$ acting transitively on an open set $U$ in a real Euclidean space of lower dimension. Given a smooth, compactly supported function $f: U \rightarrow \mathbb{R}$ ...
4
votes
0answers
45 views
Euler number of the complex of basic forms
Let $G$ be a compact Lie group and $\pi:P \to M$ a principal $G$-bundle. I would like to understand the geometry of $M$ through $P$ with the $G$-action.
I am trying to understand the Hopf bundle ...
1
vote
1answer
116 views
Kaehler form on weighted projective space
The Kaehler potential for the standard Fubini-Study Kaehler form in projective space $\mathbb{C} P^n$ is given by:
$$\log(\sum_{i=0}^n |z_i|^2)).$$
What is the analogous formula for a Kaehler ...
2
votes
0answers
52 views
DGBV algebra of symplectic manifold
Let $(M,\omega)$ be a simply connected closed symplectic manifold. Then we have the symplectic codifferential operator $d^{\star}$. Furthermore, $(\Omega^{*}(M),d,d^{\star})$ is a differential ...
6
votes
1answer
413 views
Is there a sideways-walking rolling convex body?
Let $K$ be a solid, homogenous convex body in $\mathbb{R}^3$.
Place $K$ on an inclined plane, and let it roll down the plane,
under some reasonable assumptions of friction between $K$ and
the plane, ...
3
votes
1answer
99 views
Zero currents localized along a submanifold
Let $\mathcal{D}(\mathbb{R})$ be the continuous dual of $C^\infty_c(\mathbb{R})$, the space of compactly-supported smooth functions. There is a nice characterization of distributions ...
-2
votes
0answers
40 views
Information geometry divergence [closed]
on http://en.wikipedia.org/wiki/Information_geometry
How to derive this equation. I tried but always got 0 for each item.
$$
D[\partial_i\partial_j||\cdot]= ...
0
votes
0answers
6 views
Contraction between basis vectors and basis one-forms [migrated]
Discretion: The title may be misleading, because I am not certain whether the one-forms are actually basis one-forms.
I always thought by definition, $dx^i (e_j) =\delta^i_j $.
But, I am confused ...
9
votes
2answers
312 views
The moduli space of special Lagrangian submanifolds
Given a special Lagrangian fibration $f:M \rightarrow B$ of a Calabi-Yau manifold $M$, one can associate to it two affine structures (symplectic and complex) on the base space $B$. A theorem of ...
5
votes
2answers
266 views
Can eta invariant be written in terms of topological data?
The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In ...
1
vote
1answer
77 views
Normal coordinates near the boundary
Let $M$ be an Riemannian manifold with boundary $\partial M$ and $e_n$ be a unit normal vector on $\partial M$. With respect to $e_n$, around a point $p$ on boundary, we have the usual normal ...
0
votes
0answers
41 views
Total differential of Lipschitz submanifolds embedding
My interest is analysis on Lipschitz manifolds. I want to define traces of differential forms on a Lipschitz submanifold $N$ of a Lipschitz manifold $M$. In other words, I want to push-forward ...
1
vote
1answer
189 views
Why tangent vector of statistical manifold is a function?
In differential geometry, tangent vectors are considered operators.
At point p, the local tangent space is defined as
$$
T_p(M)=\{X^i\partial_i|X\in R^n\}
$$
This is quite easy to understand for me.
...
3
votes
0answers
87 views
Moment map in the singular case
The moment map is defined on the symplectic manifold $(M,\omega)$, or particularly, $(M,\omega)$ is Kahler. While $\omega$ is smooth or differential enough, the definition is obvious to understand, in ...
2
votes
1answer
112 views
Riemannian metric and Volume form for $SE(n)$ and/or $E(n)$
I wonder what happens when you construct the Tiling spaces considering the natural action of $SE(n)$ or $E(n)$ rather than $\mathbb R^n$. In order to do that, I need to understand both the (left ...
0
votes
0answers
70 views
Exposition of the Calabi complex
I am interested in a complex derived by Eugenio Calabi in his article "On compact Riemannian manifolds with constant curvature".
The complex is referenced as "Calabi complex" in various citing ...
2
votes
3answers
184 views
Diffeomorphism with prescribed behaviour
If $\gamma$ and $\eta$ are two smooth curves in a smooth manifold $M$, is it possible to find a diffeomorphism of $M$ such that $f \circ \gamma = \eta$? What if one removes the assumption of ...
1
vote
0answers
130 views
Cotangent bundle of symmetric space is symmetric space?
Let $G$ be a connected Lie group. Then a symmetric space for $G$ is a homogeneous space $G/H$ where the stabilizer $H$ of a typical point is an open subgroup of the fixed point set of an involution ...
0
votes
0answers
106 views
Fiber bundle trivialization. Transition functions
Depending on the authors, trivialization is considered either as a diffeormorphism from $U\times G$ to $\pi^{-1}(U)$ or from $\pi^{-1}(U)$ to $U\times G$. The result leads to transition functions ...
4
votes
0answers
66 views
Is the $L^2$ metric on the space of unit volume Riemannian metrics on a closed, oriented surface Kahler?
Let $\Sigma$ be a closed, oriented, smooth surface. Denote by $\mathcal{M}^{1}(\Sigma)$ the deformation space of unit volume Riemannian metrics on $\Sigma:$ here we consider two metrics equivalent if ...
2
votes
0answers
47 views
Complete gradient shrinking Ricci soliton with nonnegative Ricci curvature?
Besides the product of a positive Einstein manifold with the Euclidean Gaussian shrinker, does there exist other complete (nonconpact) gradient shrinking Ricci soliton with nonnegative Ricci ...
3
votes
2answers
140 views
Nielsen-Thurston classification of homeomorphisms for open surfaces?
In Proposition 3.1. in this article by John Franks, he applies the Nielsen-Thurston classification of surface homeomorphisms to a homeomorphism $ \ f:M \rightarrow M$ of an open surface $M$ which is ...
5
votes
5answers
832 views
Can anyone give an example of Ricci flat Riemannian or Lorentzian Manifold that is not flat?
Does there exist a Ricci flat Riemannian or Lorentzian manifold which is geodesic complete but not flat? And is there any theorm about Ricci-flat but not flat?
I am especially interset in the case ...
10
votes
1answer
259 views
Do there exist non-totally geodesic isometric minimal immersions $\mathbb{H}^2\rightarrow G/K.$
Suppose $G$ is a non-compact, semi-simple Lie group, of rank at least two, with maximal compact subgroup $K$ and $G/K$ the corresponding Riemannian symmetric space. Let $\mathbb{H}^2(-c^2)$ be the ...
0
votes
0answers
32 views
Buseman function is regular on manifolds without boundary containing a line?
For an n-dim noncompact manifold M without boundary. Assume M contains a line $\gamma$. For every point $p \in M$, let $\widetilde{p\gamma(t)}$ be the geodesic from p to $\gamma(t)$.
Choose a ...
1
vote
1answer
323 views
why quintics are Calabi-Yau?
Why quintics are Calabi-Yau? Is there a explicit formula of the holomorphic volume form?
4
votes
1answer
219 views
Given Gaussian curvature, can one construct a metric to fulfill the Gauss-Bonnet theorem?
Consider a compact surface $\mathcal{S} \subset \mathbb{R}^3$ without boundary and assume we are given the Gaussian curvature $K(x), x\in \mathcal{S}$. It is know that Gaussian curvature does not ...
2
votes
1answer
193 views
Cotangent bundle of coadjoint orbit is stein manifold?
Let me first define stein manifolds and coadjoint orbits.
A complex manifold $X$ of complex dimension $n$ is called a Stein manifold if the following conditions hold:
$X$ is holomorphically convex, ...
1
vote
0answers
78 views
Aysmptotic comparison of L^2 sections versus generating sections
Let $s_1,\ldots, s_k$ be linearly independent global holomorphic sections of a holomorphic line bundle $E$ over a compact algebraic manifold $X$, with volume form $\Omega$.
For $m$ large, let ...
1
vote
0answers
85 views
Distributing the Hodge map over the wedge product
Let $(V,\langle,\rangle)$ be a finite dimensional inner product space, $V^{\wedge}$ it exterior algebra, and $\ast$ the Hodge star arising from $\langle,\rangle$. Does there exist any formula to ...
0
votes
0answers
56 views
Fill radius and fundamental group
I am reading M. Ramachandran and J. Wolfson's article Fill radius and fundamental group, whose main result is:
Theorem. Let $N$ be a closed Riemannian manifold. If its universal cover has fill ...
-1
votes
0answers
78 views
If there is a diffeomorphism between two surfaces, what is the relation between Laplace-Beltrami operators on the surfaces?
Let $S(0)$ and $S(t)$ be hypersurfaces of dimension $n$ in $\mathbb{R}^{n+1}$. Suppose there is a diffeomorphism
$F^0_t:S(0) \to S(t)$. Denote the Laplace-Beltrami operator by $\Delta_{S(\cdot)}$. Let ...
2
votes
1answer
115 views
Existence of planar orthogonal curvilinear coordinates on a surface embedded in $R^3$
We consider a surface (co-dimension 1) in $R^3.$ I read from the book of Stoker that for any surface there always exist patches of orthogonal curvilinear coordinates that cover the surface.
I want ...
6
votes
1answer
266 views
When a symplectic manifold is formal?
Let $(M,\omega)$ be a $2n$-dimensional symplectic manifold, then we have the symplectic Hodge operator
$$*:\Omega^{k}(M)\rightarrow\Omega^{2n-k}(M)$$
Furthermore, we can define a differential ...
