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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).

32835 questions
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Calculating the derivative of a differentiable map between manifolds

Let $M$ be a smooth manifold. Let $p\in M$. A tangent vector at $p$ is a an equivalence class $[\gamma]$ of smooth curves $\gamma : (-\epsilon,\epsilon)\rightarrow M$, with $\gamma(0) = p$, where the equivalence relation is as follows:…
Jarne Renders
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What is wrong with my proof? (A problem of tangent bundle)

I am proving $TS^1$ is diffeomorphic to $S^1\times\mathbb{R}$. The following is my proof and I think it is wrong, because I only use the fact that $S^1$ is 1-dimentional. However, I do not know how to correct my proof. ($S^1$ is the unit…
YYF
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Applying directional derivative operator to a function

I'm reading up about the more general definition of directional derivatives, but still in the context of $\mathbb{R}^n$. It goes like this: Let $f:\mathbb{R}^n\to\mathbb{R}$ be a real-valued function on the manifold $\mathbb{R}^n$ and let $v_p$ be…
Shirish Kulhari
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How to calculate the Tangent space of a line in $\mathbb{R}^2$?

Calculate the Tangent Space of a line $y=mx; m\in \mathbb{R}$ in a point $p=(x,y)$ I know that a line that passes through the origin is a manifold $M$ and the chart is $(M,\varphi); \varphi(x)=(x,mx)$. I know that $T_pM$ is generated by one…
sango
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Why do we need charts to define a submersion/immersion?

I am learning the theory of smooth manifolds and have a question on the definitions of a submersion/immersion and its dependency on given charts. Given a smooth map $f:M\mapsto N$ between two smooth manifolds of finite dimension. If I am correct…
Fermat
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Umbilical points plus isometric surface exercise

EXERCISE a)Show that there is not surface with first fundamental form: $$ds^2=udu^2+vdv^2$$ that can be isometric with a circular cylinder b)Find the Umbilical points of the elliptical…
Paris K. Patsogiannis
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Transformation of Yang Mills field

I am trying to recover the transformation properties of a Yang-Mills field and I'm not sure if I am wrong or if I am misunderstanding what is meant by a Yang-Mills field. Suppose I had a principle $G$ bundle $P \xrightarrow\pi M$ and a connection…
YankyL
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Regular surface at critical value; procedure for "find real numbers $c$ satisfying $F(x,y,z) = c$ be a regular surface. "

First Let me state what I know, If $r$ is a regular value of $F$ and $F^{-1}(r)$ is non-empty then $F^{-1}(r)$ is a regular surface Of course I know, its converse is not true. Typical example can be found in Do Carmo as $F(x,y,z) = z^2$. The…
phy_math
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Bounds on the Christoffel symbols in normal coordinates

Given a smooth closed Riemannian manifold $(M,g)$ with injectivity radius $i_M$ I can choose for every $x \in M$ an orthonormal basis of $T_x M$ and a normal coordinate system $ \phi$ in $x$, such that the Christoffel symbols…
djamba
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Why is there zero holonomy around a geodesic?

I'm reading about the Gauss-Bonnet theorem, and in the beginning of the section I'm studying I saw: Why is this true? I know this should be trivial but I'm not seeing it.
Matheus Andrade
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Critical points and immersion map

I was dealing with the following problem from differential geometry yesterday. Let $ x_1, \ldots, x_4 $ be points in general position in $ \mathbb{R} ^3 $ (that is they don't lie in a plane). Let $ q_1, \ldots, q_4 \in \mathbb{R} $ be electric…
Sota Antonino
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Asymptotic curve with negative Gauss curvature, show $|\tau(P)|=\sqrt{-K(P)}$

Suppose $K(P) < 0$ where $K(P)$ is the Gauss curvature at $P$, where $K(P) = \det|S_p|$, the determinant of the shape operator at $P$. If $C$ is an asymptotic curve with $\kappa(P) \neq 0$, prove that its torsion satisfies…
user637978
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How to find the area vector of a shape?

If I have a simple $3-d$ shape like a square plate connected to an identical square plate at one edge and they are at an angle of $90\, ^{\circ}$, how would I find an area vector that describes it? I think it might involve finding a unit vector…
Marcus
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Signed curvature of catenary involving turning/tangential angle

Suppose we want to find the signed curvature of the catenary $$\gamma(t)=(t,\cosh t)$$where $\mathcal{k}_n=\frac{d\phi}{ds}$ and $\phi(s)$ is the turning angle of $\gamma$ such that$$\dot\gamma(s)=(\cos\phi(s), \sin\phi(s))$$ We proceed:…
user573025
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A question about the definition of regular surfaces in Manfredo do Carmo's book

I have a question in Manfredo do Carmo's book: Differential geometry of curves and surfaces. According to the explanation of definition of regular surfaces, the condition 2 can avoid some kind of "self-intersection" which are shown with a figure in…
H.J. Chou
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