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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.

Differential geometry is a branch of mathematics that uses techniques in calculus and linear algebra to study geometry. It is closely related to differential topology and PDEs (geometric analysis).

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Differential Geometry: How to find curvature and torsion, given only the Binormal

I have to find the curvature and torsion of a curve (parametrised by arc length), given only the Binormal vector. Whilst I understand how to find these if I have the curve, I cannot for the life of me work out how to go in this direction. Any help…
Stephen Palmer
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Isometric expression of second fundamental form coefficients

Apart from the function $(LN-M^2)$ composed of purely second fundamental form coefficients ( or with derivatives) what are some other examples of such scalars which are isometrically invariant? EDIT1: Without basis and as a pure guess I…
Narasimham
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Reduction of the principal fibre bundle and canonical tensors

This is not really a specific question but rather a flawed argument and I am trying to spot where are the flaws, I apologize in advance. Given a n-dimensional smooth manifold $\mathcal{M}$ consider the principal frame bundle $F\mathcal{M}$, this…
Miguel Garcia
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Tangent spaces, and immersions: explanations needed

Hi could someone please explain the concept of a tangent space clearly and in simple terms? I've been really confused for ages about this. Also, the definition of an immersion follows from tangent spaces definition so I'm confused with that too. I…
Cay
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1-forms (on a manifold) with upstairs or downstairs indices

I was recently thrown into the unknown, for me, field of topology and differential geometry, so I picked up the popular Munkres book on Topology, and fair enough it's written in a clear and concise way. It explains the duality between vector fields…
Aayla Secura
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Canonical connection in $T^{*}M^{\otimes m}\otimes E$?

Suppose we are given a vector bundle $E$ over a Riemannian manifold $(M,g)$ ,let $\nabla^{E}$ be a connection in $E$ and $\nabla^{g}$ be the Levi-Civita connection on $M$, Is there a natural connection in $T^{*}M^{\otimes m}\otimes E$ associated to…
C Weid
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Neglected constant curvature difference surfaces

What are some surfaces where $ \kappa_1-\kappa_2$ is constant? On a sphere where all are umbilical points.. is a special case. For the $ \kappa_1+\kappa_2$ = constant case we have DeLaunay and Minimal surfaces. $ \kappa_1,\kappa_2$ are the…
Narasimham
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Self adjoint total covariant derivative

Suppose $V$ is a smooth vector field on a Riemannian manifold $M$ and the total derivative of $V$ is self-adjoint (as an endomorphism of $TM$) i.e. $$\left< \nabla V(X), Y \right> = \left< X, \nabla V(Y) \right>$$ for all vector fields $X,Y$. Why…
quietally
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When to use coordinate charts to restrict a differential form

I've been trying to understand differential forms but still have some parts of confusion. In particular, it is not clear to me when to use charts to restrict a differential form and when not. For example, consider $$\omega = \sum_{k=1}^{n+1} x_k…
self-learner
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a doubt on differential of a map between Manifolds

Let $F:N\rightarrow M$ be a $C^{\infty}$ map, At each point $p\in N$, the map $F$ induces a linear map of tangent spaces, called its differential at $p$, $F_{*}:T_p N\rightarrow T_{F(p)}M$ as follows. $X_p\in T_pN$ then $F_{*}(X_p)$ is the tangent…
Myshkin
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Fiber bundle $U(n-1)\rightarrow U(n)\rightarrow S^{2n-1}$

Can someone help me to visualize geometrically the fiber bundle $U(n-1)\rightarrow U(n) \rightarrow S^{2n-1}$, what are the open sets where it trivializes?
Morton
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What is area form?

Text book says: The area form $dS$ of a surface $S\subseteq \mathbb R ^3$ is defined as for any positively oriented orthonormal frame $\{E_1, E_2\}$, $dS(E_1,E_2)=1$. Then given parametrization $ x:D\rightarrow M$ where $M$ is a surface in $\mathbb…
Darae-Uri
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Ruled surface of constant K gauss curvature

Ruled surfaces have negative /zero K ; so what are some examples, with parametrization, of a ruled surface with constant negative K ? EDIT1: For standard ruled surface we need to integrate linked Reference Equn (14.11): $$ K= \dfrac{-M…
Narasimham
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When can we write evolving curves as curves over a fixed curve?

Suppose $\gamma(t)$ for each $t$ is a curve. We may write $$\gamma(t)(s) = \gamma_0(s) + d(t,s)N(s)$$ where $\gamma_0$ is some fixed curve, $N$ is the unit normal vector and $d$ is a distance from $\gamma_0$ to the curve $\gamma(t)$. Under what…
Pink Panther
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Local coordinates for a tranversal intersection of a curve with a coordinate axis

Suppose I have a smooth curve $\gamma: \mathbb{R} \rightarrow \mathbb{R}^2$ in the $xy$-plane given by $t \mapsto \gamma(t)=(\gamma_1(t),\gamma_2(t))$ which intersects the $x$-axis transversely. Is it then possible to locally express $\gamma$ in…
Novo
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