# Questions tagged [dg.differential-geometry]

Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

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### Chern class cohomology coefficients complex/real/integral? [migrated]

I am reading Chern classes from Kobayashi and Nomizu.
Given a vector bundle $\pi:E\rightarrow M$ with fibre $\mathbb{C}^r$ and Group $GL(r,\mathbb{C})$ they associate for each $k\leq r$ a cohomology ...

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132 views

### determinant of curvature (notation issue)

This is when studying about Chern classes from Kobayashi and Nomizu.
Let $\pi:E\rightarrow M$ be a complex vector bundle with fibre $\mathbb{C}^r$ and Group $G=GL(r,\mathbb{C})$.
Let $p:P\rightarrow ...

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62 views

### Some questions on defining the analytic index

The questions I have are about the definition of the analytic index of a family of self-adjoint Fredholm operators parameterized by a compact space $B$ (say a closed manifold). Actually, the ...

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63 views

### Module of Kahler differentials for manifolds [duplicate]

Let $A$ be a $k$-algebra and let $\mathcal{M}_A$ be the set of all $A$-modules. In $\mathcal{M}_A$, there exists a universal object $\Omega_{A/k}$, called the module of Kahler differentials, and a $k$...

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### convergence and a mean curvature condition imply convexity

I have a question regarding the proof of theorem 4.6 in https://arxiv.org/abs/1007.3899 (Hall's conjecture).
Let $S^2$ be the class of Borel subsets in $\mathbb{R}^2$ with finite and positive ...

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136 views

### Can a harmonic function on a topological cylinder have critical points?

Let $M$ be an oriented closed smooth manifold, and let $C=M\times[0,1]$, the cylinder over $M$. Let $g$ be an arbitrary Riemannian metric on $C$ (in particular, $g$ may look nothing like a product ...

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71 views

### Approximation argument in geometric flows

I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken and I'm stuck in the following ...

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87 views

### Do orientation preserving diffeomorphisms preserve homological intersection?

Let $M$ be a $4$- dimensional oriented manifold and let D be a 2-dimensional submanifold. Is it true that any $\phi \in \text{Diff}^+(M)$(the set of orientation preserving diffeomorphisms) preserves ...

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### Gradient estimate for Poisson equation on manifold

In Gilbarg-Trudinger's book 'Elliptic Partial Differential Equations of Second order', the maximum principle is used to derive the following gradient estimates for Poisson equations on Euclidean ...

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604 views

### Motivation for definition of Quotient stack

I am reading "Some notes on Differentiable stacks" by J. Heinloth. In that paper, the notion of quotient stack is defined as follows.
Let $G$ be a Lie group action on a manifold $X$ (left action). ...

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130 views

### Is there a Riemannian submersion from $Gl(2,\mathbb{R})$ to the Poincare half plane?

Let $\mathbb{H}$ be the Poincare half plane with the hyperbolic metric. Let $Gl(2,\mathbb{R})$ be equipped with a left invariant metric?
Is there a Riemannian submersion from $Gl(2,\mathbb{R})$ to ...

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### Invariant submanifolds tangent to isotypic subrepresentations

Let $G$ be a Lie group acting on a complex manifold $M$. Let $p$ be an isolated fixed point. Let us look at the representation of $G$ on $T_pM$. Suppose $T_pM = \bigoplus V_i^{\oplus n_i}$ where $V_i$ ...

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226 views

### What is wrong with the derivation?

Let $(M^n,g)$ be a Riemannian manifold, and $T$ a symmetric $(1,1)$-tensor field, i.e., $\langle T(X),Y\rangle = \langle X,T(Y)\rangle $. For convenience, denote $$\Delta_Tu=\sum_i\langle \nabla_{e_i}\...

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138 views

### If the total space of circle bundle over hyperbolic manifold admits Riemannain metric of non-positive sectional curvature?

If the total space of circle bundle over higher genus surface admit Riemannian metric of non-positive sectional curvature?
I wish to use the result about the question and find Leeb's work 3-...

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197 views

### Flat solvmanifolds?

I was looking for some reference on solvmanifolds and came up with a paper by A. Morgan tilted "The classification of flat solvmanifolds". I know there is a complete classification of flat manifolds ...

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### Closed Kaehler--Einstein surfaces are complex ball quotients

Let $X$ be a closed Kaehler manifold of real dimension 4 endowed with a Kaehler--Einstein metric of negative curvature. Is it true that $X$ is isomorphic, as a Kaehler manifold, to a quotient of a ...

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### Batyrev's theorem in non-algebraic case

Let $X$ and $Y$ be two bimeromorphic closed Kaehler manifolds with trivial real $c_1$. Is it true that $b_n(X)=b_n(Y)$ for $n\geq 0$?

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143 views

### Sheaves on solenoids

Let $(X_n)$ be a tower of finite covering maps of compact smooth manifolds, with $f_{s,t} : X_t\to X_s$ the maps, and $\Lambda_n := f_{n,0}^{-1}\Lambda$, with $\Lambda$ the constant abelian sheaf on $...

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78 views

### Automatic plurisubharmonicity for a non-negative function

I feel confused about a point in this very short paper. On the top of page 3, it is claimed that:
If $S$ is a totally real submanifold in a compact almost complex manifold $(X,J)$, then any function ...

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112 views

### In search of a new isometric twisting invariant $ T= \tau_1.\tau_2 $

A curved line in $\mathbb R^3 $ has properties of curvature and torsion, and, on an $ \mathbb R^2 $ possesses surface scalar properties of normal curvature and geodesic torsion $ (\kappa_n, \tau_g).$
...

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134 views

### On certain 2 dimensional foliation of $Gl(2,\mathbb{R})$ deleted by scalar matrices

We consider the following 4 dimensional open manifold $$M=Gl(2,\mathbb{R})\setminus \{\lambda I_2 \mid \lambda \in \mathbb{R}\}$$ where $I_2$ is the identity $2\times 2$ matrix.
We consider the $2$ ...

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201 views

### Minimal area of Seifert surfaces

Let $K$ be a knot smooth knot in a 3-manifold $M$ and fix a metric on $M$. Let $F$ be a orientable surface of genus $g$ with one boundary component. Then we can consider the family of all maps $\...

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### Do we have estimate like $\int_\gamma \alpha \le |\alpha| \cdot |\gamma|$?

Let $(X,g)$ be a compact smooth Riemannian manifold. It is known that $H^1(X, \mathbb R)\cong \mathrm{Hom} (\pi_1(X), \mathbb R)$, namely there is a natural pairing
$$
H^1(X) \times \pi_1(X) \to \...

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### An integral estimate in conformal geometry

Let $g_0$ be a Riemannian metric of positive scalar curvature $R_0$ on a closed 4-manifold, and for some fixed constant $C$, define the set
\begin{equation}
\mathcal{S} = \{u\in C^\infty(M): ||u||_{W^{...

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63 views

### Is a null hypersurface locally achronal?

Let (M,g) be a Lorentzian manifold. A subset A of M is achronal if it does not exist any timelike curve joining two points in A.
A subset A is locally achronal if for any point in A there is an open ...

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222 views

### Theory of surfaces in $\mathbb{R}^3$ as level sets

Is there a book that treats the classical theory of surfaces in $\mathbb{R}^3$ from the point of view of level sets of a function? I seem to remember someone telling me that such a book exists, but I ...

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147 views

### Smooth structure on the space of sections of a fiber bundle and gauge group

Let $\xi$ be a fiber bundle $F\hookrightarrow E\to B$ (where every space is smooth, T2 and second countable), let $\Gamma(\xi)$ be the space of smooth sections. We can complete $\Gamma(\xi)$ with ...

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104 views

### Does $f$ have the same minimiser as $\|\nabla f \|$ for $f$ strictly convex?

This question is migrated from MathStackExchange where it seemed to be too hard. I wonder does anyone here have any ideas?
Suppose $f: K \to \mathbb R$ is $\mathcal C^2$ and strictly convex on some ...

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93 views

### Differentiating Riemannian logarithmic map

Let $(M,g)$ be a Riemannian manifold, geodesically complete, and assume logarithms are well defined and smooth.
Let $c: I\to M $ be a smooth path in $M$, and $x\in M$. Can we say something about $$\...

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366 views

### Minimizing geodesics in incomplete Riemannian manifolds

Let $(M, g)$ be a Riemannian manifold, not necessarily complete. Let $x$ be a point in $M$, and let $r>0$ be such that the exponential map $\operatorname{exp}_x$ is defined on an open ball $B=B(0,r)...

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286 views

### Underdetermined system of linear PDEs

Let $a,b$ two smooth functions from the open square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$. In particular, assume $a(t,u)$ and $b(t,u)$ be linearly independent for all $(t,u) \in I^{2}$.
I ...

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### Are there Lorentzian complex manifolds?

Quick and simple...
Is it possible to define complex structures on Lorentzian manifolds? If so, Can you point me to some example(s)?

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312 views

### On limits of manifolds

This question should be fairly elementary. I’d just like to check I’m not missing anything.
Let $\{M_n\}_{n\ge 0}$ be an inverse system of smooth manifolds with transition maps $f_{t,s} : M_t\to M_s$,...

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### How to compute the Kahler potential of a Sasaki metric

The Question
Given Hessian manifold $M$, there is a natural Kahler structures on $TM$. Is it possible to write the Kahler potential of these in terms of the Hessian potential?
Background
To ...

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51 views

### Explicit parametrization of closed space curves of constant curvature

Joining arcs of helices it is pretty easy to obtain closed curves with constant curvature and $C^2$ regularity (see http://www.heldermann-verlag.de/jgg/jgg01_05/jgg0203.pdf). Joining arcs of Salkowski ...

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134 views

### Hodge numbers of compact Ricci-flat Kaehler manifold

Assume that $M$ is a closed connected Ricci-flat Kaehler manifold $M$ of complex dimension $n\geq 3$ with $h^{2,0}(M)=0$. Is is possible that
$h^{n, 0}(M)\neq 1$
$h^{p, 0}(M)\neq 0$ for some $0< p&...

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137 views

### $Td_p$ notation of Kotschick

In this paper, notation $Td_p$ is used without explicit definition (it is stated that it is a certain combination of Chern numbers). It is claimed that HRR theorem implies
$$
Td_p(M)=\sum_{q}(-1)^q h^{...

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198 views

### Uniqueness of a compatible Kahler-Einstein structure on a symplectic manifold?

$\require{AMScd}$
Preliminaries: Let $(X,\omega,J)$ be a closed Kahler manifold. That is, $X$ is a closed $2n$-manifold, $\omega$ is a symplectic form and $J$ is a compatible (integrable) complex ...

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153 views

### The $\frak{sl}_2$-representation on a symplectic manifold

Any symplectic manifold $(M,\omega)$ carries a representation of $\frak{sl}_2$: Define the maps
$$
L: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~~~~ \Lambda: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~...

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### Moishezon manifold vs proper complex variety

Does there exist a closed Moishezon manifold that does not have the homotopy type of the analytification of a smooth proper complex variety (I think we know that every closed Moishezon manifold is ...

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### Existence of meromorphic 2-forms over normal surface singularities

Let $(X,o)$ be an isolated normal surface singularity. Denote by $U:=X\backslash \{o\}$. I am looking for conditions on $(X,o)$ under which there exists a holomorphic section $\omega \in H^0(U, \Omega^...

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### Topology of abstract varieties over $\mathbb{C}$

What are the known restrictions on the topology of complex manifolds corresponding to analytifications of smooth proper algebraic varieties over $\mathbb{C}$? I think
they have to have non-zero $b_2$ ...

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139 views

### Moishezon manifold with vanishing $b_2$

Does there exist a closed Moishezon manifold with zero second Betti number?

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### Level Sets of Harmonic Maps

Can anybody point me in the direction of some references in which the level sets of harmonic maps between Riemannian manifolds are studied. (Sorry I am unfamiliar with the area and would like some ...

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### Is a symmetric, parallel (0,2)-tensor a metric?

I'm interested in affinely connected spaces, on which a metric is not necessarily defined, i.e. $(\mathcal{M},\Gamma)$. Since (as a physicist) my goal is to consider a generalized model of gravity, I ...

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### Geometrical regularity of the projection/normalization of a curve

Let $v:\mathbb{R} \rightarrow \mathbb{R}^3/{(0,0,0)} $ be a $C^\infty$ regular arc-length parametrization of a space curve.
W.l.o.g. let us assume $v(0)=(1,0,0)$, $v'(0)=(1,0,0)$. Let $\bar{v}$ be ...

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74 views

### Reference of generalized isometries

I'm wondering if these objects have a name or are studied. No one around me knows, so I thought to ask here.
Let $\Phi:\mathbb{R}^d\rightarrow \mathbb{R}^d$ be $C^2$-diffeomoprhism, and fix $p \geq ...

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### First eigenvalue of the spherical cap

Let $S$ be the round $n$-sphere of radius $R$ in Euclidean space, and let $r$ be the intrinsic distance from the north pole. Further, let $U(r)$ be the spherical cap of intrinsic radius r. (So $U(0)$ ...

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112 views

### Explicit KE metrics

Does there exist an explicit example of a
Ricci-flat, non-flat metric on a closed manifold?
Kaehler--Einstein, non-flat metric on a closed manifold (excluding metrics on homogeneous spaces and ...

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172 views

### unit element under map of stacks $B\mathcal{G}\rightarrow B\mathcal{H}$

Let $\mathcal{G}$ be a Lie groupoid. The target map $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ is a principal $\mathcal{G}$ bundle.
This article Orbifolds as Stacks? by Eugene Lerman calls (in page $...