# Tagged Questions

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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### Invariant connection form

Let $\pi : E \to \mathbb{R}^3$ define a vector bundle with a connection form $\nabla$ on $\mathbb{R}^3$. The text that i'm am reading then goes on to say that $\nabla$ is $\mathbb{R}$-invariant in ...
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### Is a bijective smooth function a diffeomorphism almost everywhere?

Suppose I have $f: M \rightarrow N \in C^{\infty}$ a smooth bijection between $n$-dimensional smooth manifolds. Does it have to be a diffeomorphism except for a set of measure 0? I think the proof ...
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### Property of the covariant derivative

I am learning to use the covariant derivative. In particular, I am trying to verify the expression $${\nabla}_{b}[(\nabla^a S) (\nabla_a S)] = 2 \nabla^a S \nabla_b \nabla_a S$$ for an arbitrary ...
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### Differentiation under the integral sign for volumes in higher dimensions

Consider a smooth convex/compact domain $D\subset \mathbb{R}^n$ and a smooth, concave function $F:D\to \mathbb{R}$. Then we can define the function that simply takes the volume of the upper contour ...
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### What does it mean for a tangent bundle to be parallel?

I am looking at page 8 of the paper here, just below definition 2.13. I have never come across a phrase such as '$T\mathfrak{C}\subset{TM}$ is parallel' - what does it mean for a tangent bundle to be ...
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### Sequence of closed sets (Milnor's proof)

I've got a question about a descending sequence of closed sets. Milnor writes in his book "FROM THE DIFFERENTIABLE VIEWPOINT". In his proof of Sards Theorem he wrote: Let $f:U\rightarrow \mathbb{R}^p$ ...
Let $M$ be a manifold, $G,H$ be some Lie groups, $\sigma:G\to H$ be a Lie group homomorphism, $K\subset H$ a maximal compact subgroup of $H$ and $\tilde K:=\sigma^{-1}(K)\subset G$ . Let further $\... 1answer 25 views ### Divergent Curves and Complete Manifolds I'm working on a problem in do Carmo's Riemannian geometry book (chapter 7, problem 5). He states that a divergent curve on a noncompact Riemannian manifold$M$is a curve$\alpha: [0, \infty) \to M$... 2answers 36 views ### When are embeddings into Euclidean space unique up to ambient isometry? Suppose I have a Riemannian smooth manifold$M$and a smooth isometric embedding$M \hookrightarrow \mathbb{R}^n$. Is this embedding necessarily unique up to some isometry of$\mathbb{R}^n$? If not, ... 3answers 85 views ### Differential Geometry for General Relativity I'm going to start self-studying General Relativity from Sean Caroll's Spacetime and Geometry: An Introduction to General Relativity. I'd like to have a textbook on Differential Geometry/Calculus on ... 0answers 44 views ### nef Line bundles over Kähler manifolds I am trying to understand a particular property of the first Chern class of a nef line bundle over a Kähler manifold. We know in general, let$X$be a complete complex projective variety, and$L$a ... 1answer 57 views ### Abstract algebraic definition of dual tangent spaces I know that if$(M,\mathcal{A})$is a smooth manifold, the dual tangent space at$p\in M$can be defined as $$T^*_pM=I_p/I_p^2,$$ where$I_p$is the ideal of the ring$C^\infty(M)$consisting of ... 1answer 50 views ### Topology, atlas, smooth manifold Let$X$be a set,$n\in\mathbb{N}$and$((U_i,\phi_i))_i$a family of subsets$U_i\subseteq X$with injective functions$\phi_i: U_i\to\mathbb{R}^n$, which hold the following conditions:$\...
I am given that $\hat{t}=\dfrac{\hat{x}+y'\hat{y}}{\sqrt{1+y'^2}}$ and $\hat{n}=\dfrac{y'\hat{x}-\hat{y}}{\sqrt{1+y'^2}}$ are the tangential and normal vector in frenet frame. We are considering only ...