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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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Lie derivative of the Christoffel symbol

The Lie derivative of the Christoffel symbol is $$ \mathcal{L}_\xi \Gamma^k_{ij} = \nabla_i \nabla_j \xi^k - R^k_{ijl} \xi^l \,. $$ How can one prove that? And why does it make sense, because Christoffel symbols are functions? I know that the last…
user344662
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Why can we think of the second fundamental form as a Hessian matrix?

Let $f: U \rightarrow \mathbb{R}^3$ be an immersion that parametrizes a piece of a surface, and let $(h_{ij})$ be the matrix for the second fundamental form of that surface. According to pg. 70 of the text Differential Geometry by Wolfgang Kuhnel,…
Elliott
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Second fundamental form of an implicit surface

I'm interested in expressions for the second fundamental form of an implicit surface $f(x,y,z) = 0$, in terms of the first and second derivatives $f_x, \ldots, f_{yz}$. I am not looking for the expressions of a function in Monge form, $z = h(x,y)$.…
user395788
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Characterization of the sphere

Hello I wanted to prove the following statement. Let $M$ be a compact connected surface in $\mathbb{R}^3$ such that for all $d\in S^2$ there exists a plane called $\pi_d$, such that is orthogonal to $d$ and it's a plane of symmetry of $M$, then…
J. Salieri
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Understanding definition of differentiable manifold

I am currently studying a basic course on differentiable manifolds.I have read the following definition of differentiable atlas and manifolds: Definition. Let $ \mathcal{A} = {(x_{\alpha},U_{\alpha})}_{\alpha \in A}$ be an atlas on a topological…
abhishek
  • 101
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Examples of compact negatively curved constant curvature manifold

I am looking for concrete examples of negatively curved constant curvature manifold. The only example of negatively curved constant curvature manifold is the hyperbolic plane. Are there any easy examples of such manifolds which are compact.
zach
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Compute curvature tensor from constant sectional curvature

Given sectional curvature as a constant, i.e. $\dfrac{R_m(X,Y,Y,X)}{|X|^2|Y|^2-^2} = C$, I want to compute the curvature tensors $R(X,Y)Z$ and $R_m(X,Y,Z,W)$. I believe I need to use the identity $$-6R_m(X,Y,Z,W) =…
user291678
  • 565
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"Smooth" tensor field

I am learning differential geometry and I have a question on a definition. We call a tensor field a function that to a point on the manifold ($p\in M$), we associate smoothly a tensor. But what does smoothly mean here? In the book we defined a…
StarBucK
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Strange use of differentials - is $d{\bf x} \cdot d{\bf x}$ a dot product?

If ${\bf x}(s)$ is a curve in $\mathbb{R}^3$ on a surface parameterized by its arc length $s$, and ${\bf N}$ is the surface normal at ${\bf x}$, consider the following equality (with "$\cdot$" being the dot product): \begin{align*} \frac{d{\bf…
user6932
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Linear connection and covariant derivative: help needed to clear up confusion in extension of definitions

Let $\nabla$ be a linear connection defined on the tangent bundle of a manifold $M$. We have, with $X(M)$ being the global sections module of $TM$, $$\nabla: X(M)\to \Omega^1(M)\otimes X(M)$$ We can extend it as a derivation of degree 1 on the…
brunoh
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parallel vectors along a curve

What does it mean for a vector field X(t) to be parallel along a curve, gamma(t)? and how can we show that if X(t) is parallel along gamma(t), then |X(t)| is constant? Thanks
mary
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Differential of a Map

I have the following map that embeds the Torus $T^2$ into $\mathbb{R}^3$: $$f(\theta, \phi)=(cos\theta(R+rcos(\phi)),sin\theta(R+rcos(\phi)), rsin\phi)$$ noting that $0
user218512
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Christoffel symbols and fundamental forms

How can we prove that the christoffel symbol is \[ \Gamma^k_{ij} = \frac 12 \sum_{l=1}^2 g^{kl} \left(\frac{\partial g_{il}}{\partial u^j} + \frac{\partial g_{jl}}{\partial u^i} - \frac{\partial g_{ij}}{\partial u^l}\right) \] I can think of some…
Buddy Holly
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Deriving metric from Killing field.

Finding Killing fields from the metric in a Riemannian manifold is a standard procedure written in many relevant books. My question is the reverse: How can I find a metric for a manifold whose Killing fields are known?
Nemat
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Show that $S$ is non-orientable

Let $S$ be a regular surface covered by coordinate neighborhoods $V_1$ and $V_2$. Assume that $V_1\cap V_2$ has two connected components, $W_1$, $W_2$, and that the Jacobian of the change of coordinates is positive in $W_1$ and negative in $W_2$.…
MathUser
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