Questions tagged [differential-geometry]

Question 1

I am trying to get a better grasp of how to find the basis of the tangent space. Here is one example I worked on in hopes of practicing it: Consider the chart $(U,\psi)$, the manifold $\mathcal{M} = S^...

Question 2

Let me know if this is more on-topic for physics.se (or more generally, off-topic for mathematics.se). Okay, imagine you have a volume of space with instruments measuring air pressure (and for ...

Question 3

Let $M = \{(x_{1}, x_{2}, x_{3}, x_{4}) \in \mathbb{R}^{4}: x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2} = -1\}$. Prove that $M$ is a regular submanifold of $\mathbb{R}^{4}$ with dimension 3. Compute the ...

Question 4

The standard Li Yau Estimate with a curvature term states that: Let $(M^n,g)$ be a complete Riemannian manifold with $\text{Ric}\ge -(n-1)Hg, \space H\ge0$ , and let $u(x,t)>0$ solve the heat ...

Question 5

For a smooth manifold $M$, a distribution $\xi \subseteq TM$ and a point $x \in M$, consider the following quantity, which I call the local defect of $\xi$ at $x$: \begin{equation} \text{def}_{x}(\xi) ...

Question 6

I am writing because in our first course on differential geometry in Spain, our professor has decided to use the concept of triangulation to define integrals of n-forms on differential manifolds, but ...

Question 7

Suppose we have a principal Lie group bundle $M\times G\rightarrow M$ on a Riem. manifold $M$ with associated vector bundle $V$. In case we need it let $\pi: G\rightarrow End(V)$ be a representation. ...

Question 8

How accurate is the following explanation? Outside Cartesian-type fixed-metric systems, when you move one step forward, both the basis and the components can change. If you only account for the change ...

Question 9

Let $f:M\to\mathbb{R}^k$ be smooth, where $M$ is a closed manifold of dimension $n>k$. I am thinking about the continuity properties of the map $c\mapsto\mathcal{H}^{n-k}(f^{-1}(c))$. Clearly this ...

Question 10

I read Theorem 4.25 in Lee's Smooth Manifolds book. This theorem states that a smooth map $F:M\rightarrow N$ is a smooth immersion if and only if it is locally a smooth embedding (let's ignore in this ...

Question 11

As I understand it, a color model (like the RGB model) is a way of mapping the space of all human-perceptible colors to a certain manifold -- typically, but not always, either three- or four-...

Question 12

I am currently studying R. Hamilton's proof that the volume-preserving diffeomorphisms are a tame Fréchet-Lie group. The construction of a chart is at least in part motivated by the Helmholtz-Hodge ...

Question 13

While doing some calculation I came across the following term [$G=G(x,x'),$ $x$ is independent of $x'$] $$K=\frac{\partial G(x,x')}{\partial (\frac{\partial G}{\partial x'})}$$ I tried to think of it ...

Question 14

Let M be a smooth manifold and U ⊂ M an open subset. Let X ∈ X(U) be a smooth vector field defined only on U. Prove that there exists a smooth vector field $X_2$ ∈ X(M) such that $$ X_2|_U = X, \qquad ...

Question 15

Consider $\mathbb{R}^n$ with its standard inner product and usual affine connection $\nabla^{\mathrm{eucl}}$. Let $M$ be a manifold and suppose that one is given an immersion $f:M\to\mathbb{R}^n$. At ...

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Questions tagged [differential-geometry]

Confusion regarding Tangent Basis

Determining the location of portals using air pressure

Connected components on a regular manifold

Proof on the Li -Yau Inequality

Quantifying Maximal Nonintegrability

References for triangulation of manifolds [closed]

How is the Gauge Covariant derivative defined for an arbitrary principal bundle?

Understanding why the derivative of the components of a tensor are not components of a tensor

Smooth functions with arbitrarily large regular level sets

Smooth immersion is a local embedding

What mathematical structure do the manifolds described by color models and color spaces have? [closed]

The role of the de Rham theorem in $\nabla C^\infty(X)\cong C^\infty_0(X)$

derivative of integral of a function with respect to the same function

vector field on open subset of smooth manifold can be extended to the whole manifold [closed]

Do affine connections always come from immersions in $\mathbb{R}^n$?

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