# Tagged Questions

**9**

votes

**1**answer

86 views

### Harmonic function with injective boundary conditions is an immersion?

Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are given an immersion $f:M \to \mathbb{R}^n$. (i.e $df$ is invertible at every point $p \in M$, note ...

**23**

votes

**2**answers

677 views

### Why is it true that if two 4-manifolds are homeomorphic then their squares are diffeomorphic?

Near the top of the second page of this paper, it is claimed that if two 4-manifolds $X$ and $Y$ are homeomorphic, then their squares $X \times X$ and $Y \times Y$ are diffeomorphic. Why is this true? ...

**3**

votes

**1**answer

82 views

### coisotropic action on $TS^{2n+1}$

Let $S^{2n+1}$ be the $m$-dimensional sphere in $\mathbb{C}^{n+1}$. Endow $S^{2n+1}$ with the standard metric. Let $S^1$ act by multiplication on $S^{2n+1}$. Then $S^1$ and the canonical action of ...

**2**

votes

**0**answers

83 views

### Extension of functions from geodesically convex compact sets in a Riemannian manifold

In the paper Extension operators for spaces of infinite differentiable Whitney jets (J. reine angew. Math. 602 (2007), 123—154, DOI:10.1515/crelle.2007.005) by Leonhard Frerick, a convenient condition ...

**2**

votes

**0**answers

64 views

### A problem of defining addition in a Quotient space

Let $\mathcal{C}$ be the space of all parametric curves $x:[0,1]\rightarrow \mathbb{R}^2$. Let the set of all re-parameterizations of curves is $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma ...

**2**

votes

**1**answer

91 views

### Proof about affine connections

I'm reading Nomizu & Sasaki's "Affine Differential Geometry: Geometry of Affine Immersions" and I'm having some trouble with Proposition 1.4.
I have an immersed surface in $M \hookrightarrow ...

**3**

votes

**0**answers

66 views

### Divergence theorem on stratified spaces

It is very common in physics and engineering to apply the divergence theorem
to compact spaces whose boundary is not smooth. For example, in the wikipedia link I just gave, the picture illustrating ...

**18**

votes

**6**answers

622 views

### Automorphism group of real orthogonal Lie groups

I would like to know what is the "outer-automorphism group" $Out$ of $SO(p,q)$ and $O(p,q)$, where $p+q >0$. My working definition of $Out$ is as follows:
Let us denote by $Aut(G)$ the ...

**3**

votes

**0**answers

171 views

### Canonical metric on moduli space of singular Calabi-Yau varieties

Let $\pi:X\to Y$ be a surjective holomorphic map with connected fibers
and let fibers are singular Calabi-Yau varieties (i.e. numerical dimension is zero) then is it possible to construct canonical ...

**5**

votes

**2**answers

146 views

### Is $\mathbb{P}T^*M$ a sub-Riemannian manifold if $M$ is Riemannian?

(this question is about a particular aspect of a previous question, which was not duly stressed)
Let $(M,g)$ a Riemannian $n$-dimensional manifold, and let
$$
\widetilde{M}:=\mathbb{P}T^*M
$$
be the ...

**6**

votes

**2**answers

218 views

### Index of Modified Dirac Operator

Let's say we have an oriented compact 4-d Riemannian spin manifold $(M,g)$. Everybody who's anybody has heard about the index of the Dirac operator $D: S^+\rightarrow S-$; it's the $\hat{A}$-genus, ...

**3**

votes

**1**answer

81 views

### Vector fields, diffeomorphism subgroups and lie group actions

Let $M$ be a compact smooth manifold. Since any vector field is complete we get a $1$-parameter subgroup for each vector field. Consider the following generalization:
Let $\{X_j\} \in Vect(M)$ be a ...

**2**

votes

**0**answers

47 views

### Action of orthogonal group on the free Lie algebra

This question is somewhat related and inspired by this post of professor Montgomery.
The free Lie algebra $L(V)$ generated by an $r$-dimensional vector space $V$ is, in the language of ...

**3**

votes

**0**answers

63 views

### Perburb the Monodromy of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk,(more general, Lefschetz fibration over a surface with boundary ) and let $\Omega$ be a closed 2-form on $E$ such that it ...

**9**

votes

**1**answer

115 views

### Generalized Dirac operators

So far I met three definitions of the so called generalized Dirac operator(or Dirac type operators. Everything takes place over Riemannian manifols $M$ and we have smooth hermitian vector bundle $S ...

**2**

votes

**1**answer

46 views

### A random question on the local family index theorem

A random question.
Say we have a complex vector bundle $E\to M$ that is isomorphic to a trivial bundle, and $d$ is a trivial connection on it. Let $M\to B$ be a fiber bundle with even dimensional ...

**2**

votes

**1**answer

127 views

### isometric action on the $n$-sphere

Let $S^n$ be the $n$-sphere. If $n=2k+1$ is odd, then we can identify $S^n$ as a subset of $\mathbb{C}^{k+1}$. We define the $S^1$ action on $S^n$ by multiplication, namely
$$ \Psi \colon S^1 \times ...

**5**

votes

**1**answer

211 views

### What is meant by a Lie group acting by affine transformations?

This question is a continuation of this one, where I did not receive a complete answer so am moving to MO. I am trying to understand a paper by JP Szaro that was referred to there, and in particular, ...

**5**

votes

**0**answers

165 views

### Homotopy groups of the Grassmannians

What are the homotopy groups of the oriented Grassmannian $Gr^{+}(p,q)$ (p-planes in $R^{p+q}$) $\pi_{r}(Gr^{+}(p,q))$, $r \le pq$?
Do you know any references on the web about it?

**2**

votes

**0**answers

81 views

### Willmore functional

Let $(M^2,g)$ and $(\bar{M},\bar{g})$ be two Riemannian manifolds. Suppose that $\mathcal{W}$ is the Willmore functional on the set of immersion functions from $M^2$ to $\bar{M}$.
We know that ...

**2**

votes

**0**answers

91 views

### How many linear independent vector fields can be constructed on a general manifold with $\chi(M)=0$?

We have known how many linear independent vector fields can be constructed on $S^n$:https://en.wikipedia.org/wiki/Vector_fields_on_spheres
So how many linear independent vector fields can be ...

**1**

vote

**0**answers

84 views

### Orders of zeros of section of sheaf

We have a semistable family (fibers has normal crossings and they are reduced, multp 1) $f: X \rightarrow Y,$ of complex curves over a smooth curve $Y.$ The family is smooth over the set ...

**1**

vote

**4**answers

146 views

### On the definition of fundamental vector field

I've encountered with following question while reading Morita's book "Geometry of Differential Forms" (pp.263)
Let $(P,\pi,M,G)$ be a principal G-bundle, $\mathfrak{g}$ be the Lie algebra of the Lie ...

**0**

votes

**0**answers

143 views

### Defining smooth manifolds without homeomorphisms

I would like to define smooth manifolds via atlases without homeomorphisms, by viewing it as a set (not a topological space) with maps and transition functions, by requiring the latter to be smooth ...

**7**

votes

**0**answers

159 views

### Biholomorphic neighborhoods of the boundary of Stein domains

Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...

**4**

votes

**1**answer

131 views

### Distance comparison in submanifold versus in the underlying manifold

Let $(M,g)$ be the (underlying) manifold, $(S,g|)$ be a submanifold. Let $a,b,c \in S$. It's not in general true that $d_M(a,b)\leq d_M(a,c) \implies d_S(a,b)\leq d_S(a,c)$.
QUESTION I:
The above ...

**5**

votes

**0**answers

87 views

### $H(M)$ necessarily highly non-integrable, i.e. forms contact structure?

Let$$M^{2n - 1} = \{z \in \textbf{C}^n : \textbf{h}(\textbf{z}, \textbf{z}) \equiv \textbf{z} \cdot \overline{\textbf{z}} = \textbf{1}\}$$be the unit sphere in $\textbf{C}^n$. Consider the ...

**5**

votes

**0**answers

79 views

### Spectra of Dirac operators

1. Suppose that $M$ is a spin manifold. The spin structure of $M$ is not uniquely defined: in other words, $M$ may have many
nonequivalent spin structures. For each choice of spin structure there is ...

**0**

votes

**1**answer

75 views

### When are the minimizing geodesics of a totally geodesic submanifold also minimizing in the underlying manifold? [duplicate]

Also asked here: http://math.stackexchange.com/questions/1725787/when-are-the-minimizing-geodesics-of-a-totally-geodesic-submanifold-also-minimiz
A reference on totally geodesic submanifold (TGS):
...

**1**

vote

**1**answer

93 views

### Riemannian metric on a level set of a smooth function on a manifold

Also asked here: http://math.stackexchange.com/questions/1725491/riemannian-metric-on-a-level-set-of-a-smooth-function-on-a-manifold
Let $(M,g)$ be a finite or infinite dimensional Riemannian ...

**4**

votes

**1**answer

73 views

### Locally nilpotent algebraic section of tangent bundle is complete?

Suppose $X$ is a smooth affine algebraic variety over $\mathbb{C}$ and let $V$ be an algebraic vector field (i.e. an algebraic section of the tangent bundle). If $V$ is locally nilpotent, meaning that ...

**11**

votes

**1**answer

377 views

### Differential geometric interpretation of cohomology

I'm not sure whether this is an appropriate question for this forum. I'm afraid that this is not a research level question however:
1. It's about reference request therefore the answer does not ...

**2**

votes

**1**answer

120 views

### Local form of Dirac operator

I have the following question.
Provided that $E\to M$ is vector bundle and that a Clifford module $Cl(T^*M)$ acts on $\Gamma(E)$ via Clifford multiplication $c$, the Dirac operator of this Clifford ...

**4**

votes

**1**answer

126 views

### Difference between the Laplacian and the sub-Laplacian of a Lie group

Given a Lie group $G$, what is the difference between the Laplacian $\Delta$ and the sub-Laplacian $\Delta_{sub}$ of $G$. And what are the properties that we lose when going from sub-Laplace to ...

**0**

votes

**1**answer

62 views

### Douglass integral and harmonic maps

Let $F$ be the family $f:U\to R^3$ of smooth mappings of the unit disk onto $R^3$ mapping the boundary $T$ onto a smooth Jordan curve $\Gamma\subset R^3$. Form the energy integral $$E[f]=\int_U ...

**21**

votes

**1**answer

289 views

### Is the normal bundle of a torus trivial?

Question:
Let $T^k \subseteq \mathbb{R}^n$, $ n > k$, be a smoothly embedded $k$-torus. Is its normal bundle trivial?
What about the normal bundle of $S^k \subseteq \mathbb{R}^n$, $n > k$, the ...

**1**

vote

**1**answer

78 views

### Property of Lie algebroid morphism: $\#_B\circ \Phi=d\phi\circ \#_A$?

Let $A\longrightarrow M$ and $B\longrightarrow N$ be Lie algebroids with anchors $\#_A$ and $\#_B$, respectively.
A morphism of Lie algebroids is a morphism of vector bundles $\Phi:A\longrightarrow ...

**7**

votes

**0**answers

244 views

### Can any smooth triangulation of a smooth manifold be blurred?

For the purposes of this question, let's say that a blurring
of a smooth triangulation $T$ of a smooth manifold $X$
is a smooth homotopy $h\colon [0,1] \times X \to X$ such that ...

**2**

votes

**1**answer

251 views

### Existence of non-constant solutions for this equations

This question is related to this question: "Solutions of equations characterizing a complex structure." Where, here we suppose the Euclidean space instead of Sphere and the following equations happen ...

**16**

votes

**1**answer

394 views

### Just how close can two manifolds be in the Gromov-Hausdorff distance?

Suppose that we have two compact Riemannian manifolds $(M,g)$ and $(N,h)$. Define the Gromov-Hausdorff distance between them in your favorite way, I'll use the infimum of all $\epsilon$ such that ...

**6**

votes

**2**answers

158 views

### Criterion for deciding the conformal class of a metric on a complete surface

For orientable closed Riemannian surfaces $(S,g)$, there is a constant curvature metric $\overline{g}$ on $S$ that is conformal to $g$ in the sense that $\overline{g} = e^ug$ for some smooth function ...

**4**

votes

**1**answer

259 views

### On the complexification of a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle. If we suppose $TM\otimes\mathbb{C}$ is the complexification of $TM$ then how can we define a natural metric on the complex bundle ...

**11**

votes

**1**answer

276 views

### A symmetric embedding of manifolds

Assume that $M$ is a manifold.
Is there an embedding of $M$ in some $\mathbb{R}^{n}$ such that the image of $M$ in $\mathbb{R}^{n}$ is invariant under each reflection $(x_{1},x_{2},\ldots ...

**1**

vote

**0**answers

38 views

### Continuity of Busemann-Hausdorff area density

I am trying to find out why the Busemann-Hausdorff area density as defined by Burago and Ivanov is continuous. Here, $GC_m(V)\subset \Lambda^m(V)$ denotes the simple $m$-vectors in an $n$-dimensional ...

**0**

votes

**0**answers

152 views

### About the homotopy type of diffeomorphism groups

In this paper by Antonelli, Burghelea and Kahn (Topology, 1972), a homomorphism $L :\pi_i(\operatorname{Diff}(S^n, D_+^{n})) \rightarrow \Gamma^{n+i+1}$ was used as a tool to detect non-triviality of ...

**0**

votes

**0**answers

57 views

### Integral of gradient between level sets of Lipschitz functions

Start with a compact metric measure space $(X,d)$, with a doubling measure $\mu$ and a local regular Dirichlet form $\mathcal E$ that supports a Poincare inequality. $d$ can be taken to be the ...

**13**

votes

**2**answers

240 views

### Classification of $O(2)$-bundles in terms of characteristic classes

I had asked this question in stackexchange but there seems to be no consensus in the answer
It is well-known that $SO(2)$-principal bundles over a manifold $M$ are topologically characterized by ...

**2**

votes

**2**answers

134 views

### Finding a specific Global Smooth Function

Any help with this problem would be appreciated. Thanks
Suppose $(M^3,g)$ is a smooth compact Riemannian manifold with smooth boundary and $\gamma$ is a simple smooth orientable curve in $M$. Does ...

**0**

votes

**1**answer

97 views

### Variation of normals along loops

I've constructed a quotient space $M/\sim$ in $\mathbb{R}^d$ that must be a $2$-manifold. If $M/ \sim$ is a sphere then I know that its normal spaces must vary at least 90 degrees. That is $M / \sim$ ...

**4**

votes

**1**answer

63 views

### Orientability of orbit type strata of Lie group actions

Let $G$ be a compact Lie group that acts on a smooth, finite dimensional, oriented manifold $M$, and suppose that such action preserves orientation, i.e., for each $g\in G$, the diffeomorphism $\mu_g$ ...