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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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Finding Christoffel Symbol

Given a sphere, find the christoffel symbols $X(\theta,\phi)=(r \sin\phi \cos\theta,r \sin\phi \sin\theta, r \cos \phi)$ So $$(g_{ij})=\begin{pmatrix} r^2\sin^2\phi& 0\\ 0& r^2 \end{pmatrix}$$ $$(g^{ij})=\frac{1}{r^4 \sin^2 \phi}\begin{pmatrix}…
gbox
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Problem to understand first and second fundamental form of a surface.

Question 1 Let $\Sigma$ a surface parametrized by $X(u,v)$. Than at $p=(u,v)\in \Sigma$, the tangent plan $T_p\Sigma$ is generated by $\{X_u,X_v\}$. So any vector in $T_p\Sigma$ can be written as $aX_u+bX_v$. In particular, the scalar product on…
user386627
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How would you define a geodesic in $\operatorname{SO}(2n)$ and $\operatorname{SO}(2n+1)$?

Consider me a beginner. I am trying to find the intrinsic diameter of $\operatorname{SO}(n)$ and $\operatorname{SO}(2n+1)$, but I am unsure how to define a metric between two different points in either of these spaces. When I asked my professor…
Mike Flynn
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Vector fields on the product manifold

Regarding the isomorphism $T_{(x,y)}(X \times Y)= T_xX\times T_yY$ between the tangent spaces of two manifolds $X$ and $Y$, is it true that $$\mathfrak{X}(X \times Y)=\mathfrak{X}(X) \oplus \mathfrak{X}(Y),$$ where $\mathfrak{X}(-)$ denotes the…
Minkowski
  • 1,518
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Differential in local coordinates and Jacobian matrix

I have to solve the following exercise: For a smooth map $F: M\longrightarrow N$ between manifolds, and for given $v\in T_pM$, compute the differential $dF_p(v)$ in local coordinates, using its definition as the derivation at $F(p)$ that acts on…
Gustav.G
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Given a basis $\{e_1,\ldots , e_n \}$ of $T_pX$; there exists a chart $(U;u_1,\ldots , u_n)$ such that $\frac{\partial}{\partial u_i} |_p =e_i$

I am stuck with a well-known differential geometry property I cannot show. Let $X$ be a manifold and $p \in X$. Given a basis $\{e_1,\ldots , e_n \}$ of $T_pX$; there exists a chart $(U;u_1,\ldots , u_n)$ centered at $p$ such that…
Minkowski
  • 1,518
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don't understand why the intrinsic properties of a sheet of paper and a cylinder are the same

Hello I'm not sure if I used the correct terms but imagine that we have this sheet of paper with these two points: image they are the opposite side of the paper. But if now we rolled it up into a cylinder (taking the two red points as extremities)…
Gr_Fan
  • 55
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Application of inverse function theorem geometry

Could you please explain the following statement: Let $X,Y$ be manifolds, and assume that $\bar{X}$ is compact, and let $\phi:\bar{X} \rightarrow Y$. Suppose $y_0 \in Y-\phi(\partial X)$ is a regular value of $\phi$. Then from the inverse…
Euler....IS_ALIVE
  • 4,707
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Exterior derivative well-defined

In a differential geometry book I am currently reading the exterior derivative for a k-form $\omega$ on a manifold is defined via coordinate-patches, that is given a chart $(U,x)$ and a coordinate representation of $\omega$ on…
user486170
  • 23
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Derivation of composition

Consider $I\xrightarrow{c}M\times N\xrightarrow{f}\mathbb{R}$ where $M,N$ are smooth manifolds, $f$ is smooth function, $I$ is an open interval say $(-\varepsilon,\varepsilon)$ and $c=(c_1(t),c_2(t))$ be a smooth curve on $M\times N$. I am trying to…
user312648
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Automorphism on a trivial principal bundle

I have a simple question, related to a precedent post: Invariant vector field by group action. $M$ is a n-manifold. $P = M \times U(1)$ is a trivial principal bundle over $M$. $X$ is a vector field over $P$. $\Phi_t$ is the flow of $X$. $u$ and $z$…
vkubicki
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Identity on covariant derivative

I'm looking to prove the following identity for my homework: "Denoting the Riemann curvature tensor of the first kind by $R^i_{j k l}$ and letting $\left( V^i \right)$ be a contravariant vector, prove that $ V^i_{,k l} - V^i_{, l k} = -R^i_{ r k l}…
Harry Williams
  • 1,779
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How can I find a covariant vector in different coordinates

Suppose $B$ is a covariant vector in $\mathbb{R}^{2}$ and that $B = $ in the $x^{1},x^{2}$ coordinate system. Find $B$ in the new coordinate system $\bar{x}^{1},\bar{x}^{2}$ if $$ \bar{x}^{1} = (x^{2})^{3}$$ $$ \bar{x}^{2} =…
user8290579
  • 886
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Immersion of the manifold in the Flowout Theorem

I am reading John Lee "Introduction to Smooth Manifolds." Flowout theorem states (Theorem $9.20$, Flowout Theorem): Suppose $M$ is a smooth manifold, $S\subset M$ is and embedded $k$-dimensional sumbanifold,and $V$ is a smooth vector field that is…
Vadim
  • 633
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Smooth surjective map from $M$ to $(S^1) ^n$.

This question is for an homework. It follows like this : Let $M$ be an $n$-dimensional manifold. Construct a smooth surjective map from $M$ to the torus $(S^1)^n$. My first idea was to use the smooth covering map $ \epsilon^n : \mathbb{R}^n…
Sov
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