Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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Examples of categorical adjunctions in analysis and differential geometry?

In a lot of introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis and ...
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Compact surfaces with boundary of constant negative curvature

Consider a surface (with boundary) diffeomorphic to $S^1 \times [0, 1]$ and with constant negative curvature, sitting inside $\mathbb{R}^3$. All the examples I know of such surfaces are "part of" (or ...
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General form of a connection with zero curvature

I am looking for proofs of the following two theorems: Theorem 1. On a connected and simply-connected open set $\Omega\subset\mathbb{R}^3$, functions $L^p_{ij}\in C^1(\Omega)$ are given that satisfy ...
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Some questions about synthetic differential geometry

I've been trying to read Kock's text on synthetic differential geometry but I am getting a bit confused. For example, what does it mean to "interpret set theory in a topos"? What is a model of a ...
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Basic question: Condition for a map associated to a linear series to be an immersion

I am reading this set of lectures of a class by Prof. Harris. There is a theorem. Let $X$ be a Riemann surface and $\phi:X\rightarrow\mathbb{P^r}$ be the map defined by a linear series without ...
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$K = 0$ and $K = \text{const}$ surfaces produce $k_g = \text{constant}$ intersections?

Generally speaking, when do constant K (Gauss curvature) and zero K surfaces intersect to produce lines of constant geodesic curvature $ k_g $ ? Small circles on a sphere are examples. Or more ...
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Determining an explicit line bundle over surface

The following is a explicitly defined complex line bundle $E\to\Sigma$ over a closed surface: View $\Sigma$ as a subset of $\mathbb{R}^3$ and consider its Gauss map $n:\Sigma\to S^2$ given by ...
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Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
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Integration of bundle-valued differential forms

The literature, at least textbooks, seems to be very scarce on the topic of integrating bundle-valued differential forms. So I wonder where can I read on the topic? I want to see usual theorems, like ...
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141 views

Why is $a_{n}(x,y)=a_{n}(y)$?

This particular question is connected (with a slight variation in the definition of $g$) to an earlier question. The link is here. The specifics are: Given that $u(x,y)$ is the solution of a PDE ($x$ ...
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Decomposition of Laplacian into tangental and normal components w.r.t. submanifold

If I have the covariant Laplacian operator acting on a tensor e.g. $\nabla^2 h_{\mu\nu}$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) submanifold, how can I "decompose" the ...
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universal covering of punctured plane and Poincaré metric

I want to prove the following result: Let $\Omega$ be the domain $\mathbb{C}\backslash \{a_1,a_2,a_3,a_4\}$. Its universal covering is the unit disk, and the standard Poincaré metric is pulled back to ...
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Geodesic equation from Christoffel symbols

Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the $n$-dimensional manifold of all (strictly positive) probability vectors (distributions) on $\mathcal{X}=\{x_0,\dots,x_n\}$, i.e., each ...
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68 views

Picture behind $SO(3)/SO(2)\simeq S^2$

Is there some kind of intuitive/waving hand argument to explain that $$SO(3)/SO(2) \simeq S^2 \; ?$$
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Universal Property for Tangent Space

As far as I know there are basically three different approaches to the tangent space -all of them coming with advantages and disadvantages: Though the oldest one precisely represents the derivative ...
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Does there exist a vector field s.t. all orbits are dense on $\mathbb{R}^2$

Is there a complete vector field such that the all orbits are dense on a contractible manifold? For example, $\mathbb{R}^2$,the interior of a $n$-unit ball.
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Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper ...
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General geodesics

How to solve the following: Let $f : (M,\nabla)\rightarrow (\overline{M},\overline{\nabla})$ be a diffeomorphism of manifolds with torsion-free connections. a) For reparametrisation $\alpha$ of ...
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An imcomprehensible proof on Arnold's Lectures on PDE, contact diffeomorphism

In Lectures on Partial Differential Equations, Arnold gives a theorem in the chapter Huygens' Principle in the Theory of Wave Propagation: Theorem 1 (the theory of support functions). The manifold ...
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Example of charts on $\mathbb{R}$ that are $\mathcal{C}^r$ compatible but not $\mathcal{C}^{r+1}$ compatible.

Is there a simple example of two charts where this is the case? I'm struggling to think up one.
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Is $G$ a lie group if left multiplication is smooth and multiplication is smooth near $e$?

Suppose $G$ is a smooth manifold and also a topological group. Also suppose that left multiplication $L_g : G \rightarrow G$ is smooth for any $g \in G$. Finally suppose that the multiplication map is ...
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Derivations of the algebra of differential forms

It is well known that the interior product, the Lie derivative, and the De Rham differential are derivations of the algebra of differential forms. Does there exist other derivations of this algebra ...
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A good reference for learning about super-differentiation & super-integration?

I've looked at a couple of books for basic information for super-differentiation & super-integration - Rogers Supermanifolds, and Khrennikovs Superanalysis. Unfortunately both books lack a clear ...
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approximating a variety locally by a vector space

Suppose we have $m$ homogeneous equations with integer coefficients in $n$ variables and that $m >> n$. Let $x_0 \in \mathbb{C}^n$. Question 1: is there a way to approximate the variety ...
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Is there a relation between Super Riemannian manifolds and Kähler manifolds?

(This question has a physics motivation). Could we establish any kind of relationship between Super Riemannian manifolds (super Riemannian structures on super manifolds) and Kähler manifolds, or at ...
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280 views

Second derivative of a metric in terms of the Riemann curvature tensor.

I can't see how to get the following result. Help would be appreciated! This question has to do with the Riemann curvature tensor in inertial coordinates. Such that, if I'm not wrong, (in inertial ...
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What is visualization of gradient flow of a functional?

I don't work on functional analysis but during my study, I faced gradient of a functional. I read its definition, but I can not understand why it is a useful tool? Why if a flow can be written as a ...
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464 views

Tangent bundle of a quotient by a proper action

Given a compact group $G$ acting freely on a manifold $X$, is there a "nice" way to describe the tangent bundle of the quotient $X/G$ (when it is a manifold)? In the case the group $G$ is finite, or ...
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Clarification in a paper

This is regarding a clarification in page 384 of a paper published in Annals of Statistics by Amari. In page no. 384, he defines $$R_i(t)=\frac{\partial}{\partial \theta_i} ...
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how to cover a manifold almost everywhere?

I pose the following question: Given a connected (maybe we need compactness) manifold or regular surface, can we find a single parametrization $\chi:B\to M$ from the open ball to the manifold, such ...
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182 views

Does every non-compact manifold admit an incomplete vector field?

I know that every vector field on a compact manifold is complete. The question of whether every non-compact manifold admits an incomplete vector field seems to follow naturally. I'd hazard a guess ...
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Complex vector bundles with real transition functions

After doing a bit of playing around (I think) I was able to show that the map $\operatorname{id}\otimes\ \psi : \Omega^{p,q}(X, E) \to \Omega^{p,q}(X, \bar{E})$, where $\psi $ is the conjugation map ...
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How difficult is it to impose a differential structure on a fractal?

Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply ...
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Self Intersection and Euler characteristic

Reading the "Differential Topology" of V.Guillemin and A.Pollack, i found a definition of the Euler Characteristic different from the other one using the simplicial complex and betti number (ex. for ...
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understanding this differential operator on a tensor product

I am currently trying to read the T. Kotake's paper "An Analytical Proof of the Classical Riemann Roch Theorem", in which he defines a differential operator which acts on smooth sections of a tensored ...
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317 views

Definition of Coadjoint representation for Lie algebras

I have trouble understanding the definition of the coadjoint representation of a Lie algebra. Typically you first define a natural pairing between the Lie algebra and Lie coalgebra: \begin{equation} ...
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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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Differential geometry text with categorial flavor

I am blown away by exposition of analytic manifolds in Serre's Lie algebras and Lie groups, I want more! Is there a text that treats classic topics of differential geometry like connections, ...
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184 views

How does one determine $n$-spheres of curvature?

I am aware of circles of curvature and I am simply wondering to what extent does this generalize to $n$-dimensions. Specifically, if some surface in $n$-dimensional space is represented ...
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What is the Gauss part of Gauss-Manin connection?

The definition of Gauss-Manin connection involves de Rham cohomology. Surely, Gauss didn't work with de Rham cohomology as we know it. So, what was the context in which Gauss came up with this idea?
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An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
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Co-Area formula in Riemannian geometry

I wonder if the following holds true: Let $z:[0,1]\times B^{n-1}_r(0)\to(M^n,g), (t,p)\mapsto z(t,p)$ be a diffeomorphism a.e. onto its image with respect to the Lebesgue measure on $A:=[0,1]\times ...
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Is there a method of knowing which of the Christoffel symbols of second kind survive and vanish for a given metric?

I'm trying to solve a problem and the given $g_{ij}$ metric is $$\left[\begin{array}{cc}1&0&0\\0&(x^1)^2&0\\0&0&(x^1 \sin x^2)^2\end{array}\right]$$ The non-zero Christoffel ...
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Intuition for basic concepts in differential geometry

Im studying a basic differential geometry course this semester. In the class, we defined the concept of covariant derivative (connection) as a function which takes 2 vectors fields into a vector field ...
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What properties do isospectral Riemannian manifolds share?

I'm studying the Laplacian on (compact) Riemannian manifolds, and it turns out that if the Laplacian operators of two such spaces share their spectrum (the spaces are then called isospectral), then ...
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Relationship between functional analysis and differential geometry

I am taking courses on functional analysis (through Coursera.com) and differential geometry (textbook author : O'neil) on my university. I made the following table on my own. Are the similar ...
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The missing vector derivative operation

Generally, the differential operator $d$ takes differential $k$-forms to $(k+1)$-forms. In $\mathbb{R}^n$ (or more generally on an oriented Riemannian manifold), we can identify vector fields with ...
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Is the spectrum of a first order PDO always unbounded from both sides?

Let $E \to X$ be a smooth vector bundle over a compact Riemannian manifold $X$ and assume that $P:\Gamma(E) \to \Gamma(E)$ is a self-adjoint partial differential operator of order $1$. We think of ...
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Why not differentiable manifolds that are not of class $C^1$

In most, if not all (I cannot say for sure) references on manifolds, we seem to consider $C^k$-manifolds, including the case $k = 0$, which corresponds to topological manifolds. This means that we ...
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Classification of Geometry

I'm asking for big picture in geometry here. I've studied the first three chapters of John Lee's smooth manifolds but still I cannot see the path ahead. My questions are mainly about classification of ...