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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how we can compute $\hat{i}(\alpha,\beta)$ and ${i}(\alpha,\beta)$ for following curve?

in the Farb and Margalit: A primer on MCGs. on page 28 we have : There are two natural ways to count the number of intersection points between two simple closed curves in a surface: signed and ...
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Approximate a convex body with a domain with smooth boundaries?

Let $K$ be a convex body in $\mathbb R^n$ (i.e. a set that is compact and convex). Is it possible to construct a domain with smooth boundaries that contains $K$ and is close enough in the sense of ...
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how the angle between vectors changes in a topological equivalence

I'm studying dynamical systems and i have the next problem: I have two dynamical systems in continuous time, let's say $\dot{x} =X(x)$ and $\dot{y} =Y(y)$ where there is a topological equivalence (or ...
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Do unique horizontal lifts of path homotopic paths through a point are always path homotopic?

Let $\pi:E \rightarrow M$ be a principal $G$-bundle for a Lie group $G$. Let $\omega$ be a connection on the principal bundle. It is a well known fact, that if a path $\gamma:[0,1] \rightarrow M$ is ...
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What is the "gauge field" on the base space in a gauge field theory?

Suppose we have a principal $G$-bundle $P\xrightarrow{\pi} M$, and we want to consider a classical gauge field theory (with a field Lagrangian) on $M$ for this bundle, in the physics sense. In the ...
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Calculating the product of the Riemannian manifolds and the Riemann curvature [closed]

I am a physics Master student currently taking a course on Riemannian geometry. In the course we are supposed to solve problem 7-4 out of Lee's Book Introduction to Riemannian Manifolds. The Problem ...
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Does tautological line bundle $\mathcal{O}(-1)$ over complex projective space have zero section?

In Complex Geometry written by Huybrechts, it defines the tautological line bundle $\mathcal{O}(-1)$ as followed $$\mathcal{O}(-1):=\{(\ell,z)\in \mathbb{C}P^n \times \mathbb{C}^{n+1}: z\in \ell\}$$ ...
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Weak differentiability of SDEs on manifolds

For a SDE on Euclidean spaces, it is a well known result that solutions of such SDE is smooth (with respect to Malliavin derivative on the path space of the Euclidean space) when the coefficients are ...
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Is there a distribution not expressible as a kernel of a list of 1-forms?

Given a smooth manifold $X$ and a smooth distribution/vector-subbundle $E\subset TX$ on $X$ of codimension $k$, is it possible that there do not exist $\omega_1,\ldots,\omega_k\in \Omega^1(X)$ such ...
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Does the Differential Topology/Geometry frameworks being able to model solutions to diff. eqs. that are Non-Smooth?

I don't have much knowledge about Differential Topology neither Differential Geometry, but working on this another question about solutions to differential equations, and someone recommend me to ...
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Straight line in projective plane is plane in vector space (Road to Reality , page-343)

How do we construct an n-dimensional projective space $P^n$? The most immediate way is to take an $(n + 1)$-dimensional vector space $V^{n+1}$, and regard our space $P^n$ as the space of the 1-...
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Gauss Bonnet for locally outermost closed minimal surface in Hawkings Theorem

So I'm trying to understand the proof for the following theorem from Lan-Hsuan Huang: "Trapped Surfaces, Topology of Black Holes, and the Positive Mass Theorem": Any orientable locally ...
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Are there tensor structures other than a metric which could be defined on a manifold which imply a connection through compatibility criterion?

If we say our connection is torsion free, then the metric compatibility condition completely determines it. While this is geometrically intuitive way to do it, are there other interesting tensor ...
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Can mathematics distinguish left and right?

Imagine, a mathematician from another galaxy lands on the earth. Is there a way we can explain to him what is "counterclockwise" without showing him a picture? Things like Green's formula, ...
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Which one should I start studying - Lie Groups/Algebras or Functional Analysis or Differential Geometry? [closed]

I major in Physics(an excuse for sloppy math practices) and wish to go deeper in terms of understanding the concepts and improving mathematical rigorousness. I wish to study Algebraic Topology, ...
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definition of lie bracket of linear connection and a $\operatorname{End}(E)$-valued $k$-form

I'm currently studying LECTURES ON CHERN-WElL THEORY, in which it defines the trace function on a vector bundle $(M,E)$ by setting  \operatorname{tr}(\omega\otimes A) = (\operatorname{tr}A)\omega ,\...
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Extrinsic curvature using different coordinates

I have some clarifications about the calculation of the normal vector and extrinsic curvature of a cylinder on pp. 25-26 (gr-qc/0703035), The Euclidean metric in cylindrical coordinates $(r,\phi,z)$ ...
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Can a manifold be reconstructed from its charts?

I'm learning special relativity and I am having a confusion on this mathematical point. Whenever any sort of motion or non motion happens in the world, it can only be perceived by the scientist in a ...
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In the proof of corollary associated to the closed subgroup theorem

I'am reading the John M. Lee, Introduction to Smooth manifolds and I have a question. Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. And let $\operatorname{exp} : \mathfrak{g} \to G$ be the ...
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Is the map $T_X |_S (p) := \exp{X(p)}$ a diffeomorphism onto its image?
Preliminaries The exponential map $\exp : TM \rightarrow M$ is defined by $\exp{(v)} = \gamma_v (1)$ where $\gamma_v$ denotes the geodesic starting at $p \in M$ and initial velocity $v \in T_p M$. ...