Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

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Why is a PDE a submanifold (and not just a subset)?

I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold. Let $\pi: E \to M$ be a smooth locally trivial fibre bundle. In Gromovs words a partial differential ...
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1answer
2k views

$C^{k}$-manifolds: how and why?

First of all, I have a specific question. Suppose $M$ is an $m$-dimensional $C^k$-manifold, for $1 \leq k < \infty$. Is the tangent space to a point defined as the space of $C^k$ derivations on the ...
21
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1answer
791 views

Manifolds with volume forms on every submanifold

If we equip a manifold with an inner product (i.e. we have a Riemannian Manifold) then we get a canonical volume form on that manifold (please mentally insert the prefix "pseudo" into my question ...
20
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1answer
522 views

Kähler Geodesics

Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric $$\begin{pmatrix} \frac{1}{1-|a|^2}&\frac{1}{1-a\bar{b}}\\\frac{1}{1-\bar{a}b}&\frac{1}{1-|b|^2}\end{...
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A short question on shriek maps

This should be easy but I don't quite see it. Let $M^m, N^n, X^d$ be compact, connected and oriented smooth manifolds. Let also $f:M\rightarrow X$ and $g:N\rightarrow X$ be transverse smooth maps. ...
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Interpreting the scalar curvature in a semi-Riemannian manifold

Background: Let $M$ be a smooth Riemannian manifold of dimension $n$ and scalar curvature $R$ (with respect to the Levi-Civita connection). Let $m \in M$ and let $B$ be the geodesic ball of radius $...
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441 views

Vector valued 2-forms which satisfy Jacobi Identity

Motivated by this MO question we ask the following two questions: 1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each ...
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295 views

Extension of Vector Field in the $\mathcal{C}^r$ topology.

Let $M\subset \mathbb{R}^n$ be a compact smooth manifold embedded in $\mathbb{R}^n$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is smooth and }\ X(p) \in T_p M \subset \mathbb{R}^...
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Higher-order derivatives in manifolds

If $E, F$ are real finite dimensional vector spaces and $\mu\colon E \to F$, we can speak of a (total) derivative of $\mu$ in Fréchet sense: $D\mu$, if it exists, is the unique mapping from $E$ to $L(...
15
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What is the essential difference between classical and quantum information geometry?

This question may be a little subjective, but I would like to understand, from a geometric perspective, how the structure of quantum theory differs from that of classical probability theory. I have a ...
15
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565 views

Proof of Reeb's theorem without using Morse Lemma

I'm trying to prove Reeb's theorem as stated in Milnor's Morse Theory. That is, suppose we have an $n$-manifold $M$ together with a smooth function $f$ with exactly two critical points (both non-...
14
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195 views

Show That A Particle In A Bounded Force Field Can Reach Any Point In Fixed Time Span

I tried to proof that for a smooth bounded force field $F$ and $x\in{\bf R}^n$ there exists some $v\in{\bf R}^n$ such that a particle starting in $0$ with mass $1$ and velocity $v$, obeying Newton's ...
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825 views

The heat kernel as a distance metric on manifolds

I recently came across Varadhan's formula (see e.g. [1], [2], [3], [4], [5]): $$ {d_{\text{g}}(x,y)^2}{} = -\lim_{t \rightarrow 0} 4 t \log K_t(x,y) $$ where $d_\text{g}$ is the geodesic distance ...
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Mean value operator on Riemannian manifold

Let $(M,g)$ a Riemannian manifold. Further $M$ should be a harmonic space, that is $M$ is a symmetric and simply connected space of rank 1. (Example: Spheres $S^n$) Consider the mean value operator, ...
14
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1answer
351 views

Complement of a foliation

I have an $n$-manifold $M$ which is foliated by leaves $F_\alpha$ of dimension $p$ and a path $\gamma:[0,1]\to M$. You can take without problems $\gamma$ to be injective. Is the following statement ...
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316 views

Any diffeomorphism can be locally factorised with several primitive diffeomorphisms.

ACHTUNG I ask in this question how to solve an exercise of the text Analysis on Manifolds by James Munkres: the exercise consist to prove a theorem about diffeomorphism with more restrictive ...
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Integration of a $k$-form over chains

In Spivak's Calculus on Manifolds, he defines the integral of a $k$-form over a $k$-chain, and proves a version of Stokes' theorem for this situation, before moving on to discuss the integral of a ...
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Polynomial in the components of the curvature tensor

Consider a closed Riemannian manifold $(M,g)$ of dimension n and let $K(t,x,y)$ be its heat kernel. Then it is known that the heat kernel has an asymptotic expansion as $t\downarrow 0$: $$K(t,x,x)\...
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Exponential map is surjective for compact connected Lie group

How do I show that for every compact connected group $G$, the exponential map $\exp \colon\mathfrak{g} \rightarrow G$ is surjective? I tried to find the proof on the internet but most of them are ...
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The invariance of the Ricci tensor under diffeomorphisms and its non-ellipticity.

Consider $(M,g)$ a compact Riemannian manifold. When viewed as a second order (non-linear) differential operator $$ \text{Ric} : C^{\infty}(\text{Sym}^2_+T^*M) \to C^{\infty}(\text{Sym}^2T^*M), $$ the ...
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Different notions of isometry for Riemannian $2$-manifolds

There are two notions of isometry between Riemannian $2$-manifolds: a distance-preserving map $f$ with $d(x,y) = d(f(x),f(y))$ and a "metric-preserving" map $f$ with $I(x) = I(f(x))$ ($I(x)$ being ...
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Is there a volume-preserving diffeomorphism of the disk with prescribed singular values?

This question has now been cross-posted at mathoverflow. While working on a variational problem, I have reached to the following question. Let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1\sigma_2=...
12
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1answer
266 views

Are nearby simple closed geodesics ambient isotopic?

Let $(M,g)$ be a closed Riemannian manifold. I want to show that there exists a sufficiently small $\delta > 0$ such that if $\gamma_1: S^1 \to M$ and $\gamma_2: S^1 \to M$ are simple closed ...
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Constructing $\mathbb{P^n}$ “bundle” with different $n$

Can we find a complex/smooth manifold and map $f\colon X\to\mathbb{C}^3$ such that it is a $\mathbb{P}_\mathbb{C}^m$ bundle over linear subspace $\mathbb{C}^1$ and $\mathbb{P}_\mathbb{C}^n$ bundle ...
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Determinant of non-square Jacobian

Suppose I have a 3d solid in ${\bf R}^4$ which can be parametrized by the function $F:W\subset{\bf R}^3\rightarrow{\bf R}^4$. Now suppose I want to calculate the volume of this solid. Then naively I ...
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712 views

The covering manifold of a complete Riemannian manifold is complete

This question is from Lectures on the Geometry of Manifolds by Nicolaescu. (Exercise 6.2.8 b) Let $(M,g)$ be a complete (connected) Riemannian manifold and let $(\tilde{M},\tilde{g})$ be its ...
12
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1answer
327 views

On the variation of a Kähler metric on a surface by pullback of the complex structure

Let $\Sigma$ be a compact, connected, oriented surface, and let $\rho\in\Omega^2(\Sigma)$ be a fixed volume form. Then any (almost) complex structure $J\in\Omega^0(M;\operatorname{End}TM)$ compatible ...
12
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1answer
355 views

Domain invariance for smooth functions

The domain invariance theorem states that for an open set $U\subset \mathbb{R}^n$ and a continuous and injective mapping $f:U\to \mathbb{R}^n,$ the image $f(U)\subset \mathbb{R}^n$ is open. I've read ...
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632 views

De Rham Cohomology of $M \times \mathbb{S}^1$

Let $M$ be a closed (compact, without boundary) $m$-dimensional manifold. I want to prove that $H^{k+1}(M \times \mathbb{S}^1) = H^k(M) \oplus H^{k+1}(M)$. ($H^k$ is the $k$-th De Rham cohomology ...
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“Natural” constructions of tensor fields from tensor fields on a manifold

This question begins is related to this question on physics.SE Uniqueness of Riemann Curvature Tensor, which asks roughly "what tensors can we make locally out of just the metric tensor? We can ...
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868 views

Properties of the projection onto a nonconvex set

Consider a set $\Omega\subseteq\mathbb{R}^n$ being "sufficiently regular", for example being the image of a $C^1$ mapping from $\mathbb{R}^p$ for some $p\ge1$. We may then consider the mapping $$ g:\...
12
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1answer
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A space more fundamental than Euclidean space

Summary: The mathematical physicist Paolo Budinich attributes to Élie Cartan the statement that the geometry of pure spinors is "more elementary" or more "fundamental" than Euclidean geometry, which ...
11
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Proving $\partial ^ 2 = 0 $ for the case of Morse-Complex with $\mathbb{Z}$ using orientation of the moduli space

I was going through the book Morse theory and Floer homology by Audin-Damian and got stuck where they talk about defining the complex for $\mathbb{Z}$ coefficient. Assume that $a,b,c$ are critical ...
11
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1answer
280 views

Reference Request: Equivariant cup product in singular cohomology

I've been looking around for a standard treatment of what I think sould be called "equivariant cup product in singular cohomology", but couldn't find anything promissing. I did played around with ...
11
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Is a linear vector field a geodesible vector field?

Assume that $A\in M_n(\mathbb{R})$ is a non singular matrix. Is the flow of linear vector field $X'=AX$ a geodesible flow on $\mathbb{R}^n \setminus \{0\}$?Namely, is there a Riemannian metric ...
11
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579 views

Every vector bundle has a metric connection?

Let $(E,g)$ be a vector bundle with a metric over a manifold $M$. Does $(E,g)$ always admit a compatible (metric) connection? If so, are there examples where there exists only one such metric ...
11
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Solving PDE on manifold via Hodge theory

Let $(M, g)$ be a Riemannian manifold, where $M$ is compact without boundary. The Hodge decomposition tells us that $$\Omega^k = \ker (\Delta) + \text{Im} \ d + \text{Im}\ d^* . $$ Note that we can ...
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A property processed by a special vector field.

Let $Y$ be a vector field on $\mathbb R^n$ (or any Riemannian manifold $(M, g)$). When will we know that $$Y =\nabla_X X$$ for some other vector fields $X$? Or more precisely, if $Y$ does satisfy ...
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algebraic $1$-forms vs analytic $1$-forms

First let's fix some definitions: Definitions: Complex manifold (of dimension n): Is a locally ringed space $(X,\mathscr F)$, where there is an open cover $\bigcup_{i\in I} U_i=X$ such that $(U_i,\...
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531 views

Group actions and associated bundles

Let $P$ be a principal $G$-bundle over $B$, and let $G$ act on some space $F$ (feel free to work in your favorite category of spaces, if this helps). Then $\text{Aut}{P}$ (aka the group of gauge ...
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262 views

Curvature $0$ and involutive horizontal distributions

I am trying to check why curvature $0$ implies that the horizontal distribution is involutive. Let $\pi:P\to U$ be a principal $G:=GL_n$ bundle. Assume that $P$ is trivial and $\pi$ admits a section. ...
11
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1answer
406 views

Invariant submanifolds

Let $M$ be a smooth manifold, and let $N$ be a submanifold. Let $V$ be a smooth vector field on $M$ which generates a flow $\Phi_t$ on $M$. My intuition tells me (perhaps modulo some technical ...
10
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159 views

Want to study Graduate Measure Theory with heavy Emphasis on Topology and/or Geometry.

I did one course in Measure Theory and want to study it again. But this time I want to do this in a way that emphasizes Measure Theoretic structure on Geometric or Topological Spaces. I don't know, if ...
10
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0answers
533 views

Constant Rank Theorem for Manifolds with Boundary

I'm trying to answer problem 4-3 from Lee's Introduction to Smooth Manifolds, 2nd edition. The problem says: Formulate and prove a version of the rank theorem for a map of constant rank whose ...
10
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367 views

Find sequential orthographic projections, linking three different manifolds of dimension $n=1,2,3$

Below is an image of the family of 2D curves for reference. The image is arbitrary. Still trying to formulate a concise question. I know that the geodesics for flat Euclidean Space are straight lines. ...
10
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295 views

Relation between de Rham Cohomology group of Lie group as a manifold and group cohomology of Lie group

Is there some relation between De Rham Cohomology group of Lie group as a manifold and group cohomology of Lie group? At first glance, they are two different things. De Rham Cohomology group is ...
10
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1answer
168 views

Galois covering induces an isomorphism on the level of (co)homology

The setting is the following : we have a smooth Galois cover of manifolds $p : Y \to X$, with (Galois) automorphism group $G$. Denote by $\Omega^*(X)$ and $\Omega^*(Y)$ the spaces of differential ...
10
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131 views

How weird can the boundary be so that the fundamental theorems of vector calculus hold?

Let $\Omega$ be a connected open set in $\Bbb R^n$. Suppose that I want theorems in multivariables calculus like divergence theorem or its relative like Green's identities or even Stoke's theorem to ...
10
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0answers
365 views

Gauss-Bonnet theorem proof considering membrane Force and hydrostatic fluid Pressure equilibrium

Is it possible to prove Gauss-Bonnet Theorem by using physics (Mechanics of materials) models? For example in mechanics could one consider static equilibrium by action of hydrostatic pressure ...
10
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0answers
715 views

When is a vector field hamiltonian with respect to some symplectic form?

Given a vector field $v$ on a $2n$-dimensional manifold, how many symplectic forms are there on $M$ that make $v$ a hamiltonian vector field? Alternatively, take the set of all $(H,\omega)$ pairs, mod ...

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