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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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Is this an Accurate General description of Line Bundles?

As a newbie to vector bundles, it seems like all vector bundles I have run into ( not that many, I admit) need only two charts to be trivialized; one of these charts will contain the "trouble point" (the point that prevents the top space from …
user6421
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Christoffel symbols and their transformation law

I have a problem with derivation of the transformation law for Christoffel symbols: two different approaches give me two different results. I assume that the equation for the covariant derivative of a vector shall be transformed as a tensor and…
user108687
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Closed connected integral submanifold is maximal

I'm having some problems to prove the following assertion: Let $\mathscr{D}$ be an involutive distribution of dimension $k$ in a manifold $N$. Let $(M,\varphi)$ be a integral connected submanifold, such that $\varphi(M)\subseteq N$ is a closed…
Luis Ferroni
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Maurer-Cartan 1- form as a connection 1-form

I'm trying to decipher a differential geometric comment on page 23-24 of Berline, Getzler, and Vergne's "Heat Kernels and Dirac Operators". Take a trivial vector bundle $E \times M$ in a manifold $M$ with connection $\nabla = d + \omega$ where…
Paul Siegel
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Can linear connections other than Levi-Civita connections be useful?

Consider a smooth Riemann manifold such that a Levi-Civita connection is defined. I am wondering whether there are examples in mathematics or physics where the use of other linear connections is useful despite the fact that a Levi-Civita connection…
alexl
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The connection in terms of local trivialization

I am having trouble to understand how one may write down the connection in terms of local trivialization of the vector bundle. Assume $\pi: E\rightarrow X$ is a vector bundle with rank $n$. A connection on $E$ is a map $$A:TE\rightarrow \pi^{*}E$$…
Bombyx mori
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Derivative of vector field

I have just started learning differential geometry and I am trying to answer the following exercise (it's not for homework): Let $M$ be an $n$-dimensional manifold with chart $(U, \varphi_U)$ and let $(x^1, \dots, x^n)$ be the corresponding local…
rt93
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Need more help with differential forms

The standard contact form on the sphere $S^{2n +1}$ in $\mathbb R^{2n + 2}$ is given by $$ \omega = \sum_{k=1}^{n+1} x_k dy_k - y_k dx_k$$ (see e.g. here) Now what I'm confused about is that this form uses all $2n + 2$ coordinates of $\mathbb R^{2n…
self-learner
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existence of closed surface having only negative Gaussian curvature.

I heard a theorem in differential geometry course. State of the theorem is "There is no closed (regular) surface having only negative Gaussian curvature." I tried to prove the theorem using Gauss-Bonnet theorem, but coudn't have any progress. How…
PPPiRi
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Tangential component of normal vector parallel along curve iff curve is geodesic?

Exercise 6.3 (Millman & Parker, Elements of Differential Geometry). Let $$X_N = N - \langle N, n \rangle n $$ be the tangential component of the normal vector $N$ of a unit speed curve $\gamma$ on a surface $M$. Prove that the following are…
Alain
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Zero sections of any smooth vector bundle is smooth?

Could any one give me hint how to show that the zero section of any smooth vector bundle is smooth? Zero section is a map $\xi:M\rightarrow E$ defined by $$\xi(p)=0\qquad\forall p\in M.$$
Myshkin
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Infinite surface area

I am reading an article (reference: http://www.jstor.org/stable/1971139?seq=1#page_scan_tab_contents), and in the proof of the main theorem, the author states that "it is a fact that complete, simply-connected surfaces in $\mathbb{R}^3$ of…
user1234
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The curvature is equal to the derivative of the angle between the curve and the x-axis?

I'm trying to prove that if $\vec{x}:I\rightarrow\mathbb{R}^2$ is a curve parametrized by arc length and $\theta(t)$ is the angle between the tangent line to $\vec{x}$ at point $t$ and the $x$ axis, then $\kappa=\theta'$, where $\kappa$ denotes the…
Fawcett512
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geodesic on an ellipsoid

Find all the geodesics which pass through the point $(a, 0, 0)$ on the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$. What parametrization can i use to get the first fundamental form? and how to continue after calculating E,F and G?
user333569
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Deriving the round metric

I want to derive the round metric $g=d\theta^{\,2}+\sin\left(\theta\right)^2d\phi^{\,2}$ but I cannot get the correct answer. I know that the metric in cartesian coordinates is $g=dx^2+dy^2$. I've used the formula…
user331373
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