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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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$C^k$ manifold : question about the definition

In my book they define a manifold to be of class $C^k$ if all the map linking one chart to another are $C^k$ function. But I don't really understand this as the fact that a manifold is of class $C^k$ should'nt depend on the charts we use. So do I…
StarBucK
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Generalization of Gaussian Curvature?

Consider a 2 dimensional manifold M parametrized by coordinates (x,y) embedded in $\mathbb{R}^{3}$. Thee is a smooth curve in the manifold given by $(\gamma_{1}(t),\gamma_{2}(t))$ with $t\in\mathbb{R}$. There is a smooth function $\Phi$ : $M \to…
MrLee
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Does this first fundamental form imply a surface of revolution?

A surface of revolution has the following first fundamental form $$ \begin{pmatrix} r^2(s)& 0\\ 0 &1\end{pmatrix}. $$ But does this first fundamental form imply that the surface is a surface of revolution?
gbox
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Split exact sequence of vector bundles

I have read that a short exact sequence of differentiable vector bundles is always split. I was interpreting this as the splitting being fiber wise, i.e that the fibers of the middle component of the short exact sequence are isomorphic as a…
user7090
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differential of a section, linear connections and Leibniz rule

Let $\pi:E \rightarrow M$ be a vector bundle and let $s$ $\in$ $\Gamma(E)$ (a section of the bundle). If we have a linear Ehresmann connection on the bundle, we can define a linear connection as $$\nabla s:= \rho_V \circ ds,$$ where $\rho_V$ is…
AmSa
  • 111
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Is there a "global" definition of differential $1$-forms?

Let $M$ denote a smooth manifold. Then a covector at $p \in M$ is an element of the dual space of $T_p M$. We can organize covectors into a bundle over $M$, and then define a $1$-form on $M$ to be a section of this bundle. Question. Is there a more…
goblin GONE
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Finding the instantaneous curvature of a trajectory on a 2D surface embedded in ordinary Euclidean 3D space

I have an unknown trajectory of a particle, ${\bf r}(t)$ constrained to follow a 2D surface parameterised in $\theta$ and $\varphi$: ${\bf r}(\theta(t),\varphi(t))$. I do know the surface completely (i.e. I have explicit expressions for…
robotopia
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Is the differential form of square's area as viable as circle's area?

If I want an area differential of a circle, it goes: $$dA=d(\pi \ r^2)=2\pi r dr$$ This is very useful. But What about a square? $$dA=d(l^2)=2l\cdot\ dl$$ Is this valid? Well, if I simply integrate it, of course it gives me the square's area value,…
Vitor Aguiar
  • 427
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Smooth frame for the tangent bundle $TM$

Suppose that $e_1, \ldots, e_n$ is a smooth frame for the tangent bundle $TM$ of an $n$-dimensional manifold $M$. Then, at each $p\in M$, $\lbrace e_i(p) \rbrace$ is a basis for the tangent space $T_pM$. Being a tangent vector at $p$, $[e_i,e_j]_p$…
Hussein Eid
  • 497
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Vector fields that have the same direction

Let $M$ be an $n$-dimensional $C^{\infty}$ manifold. Two vector fields $X$ and $Y$ on $M$ that never vanish is said to have the same direction if their tangent vectors $X_p$ and $Y_p$ have the same direction for each $p\in M$. Can anyone kindly…
Hussein Eid
  • 497
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Killing vector field and symmetric tensor field.

Can you tell me where I can find a proof of the following fact: Let $M$ be a Riemannian manifold and $T$ be a symmetric tensor field on $M$. Furthermore let $X$ be a Killing vector field on $M$. Then the covector field $T(X,.)$ is divergence-free.
Niklas
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List of minimal surfaces embedded in the 3-sphere

Is the set of possible areas of closed, embedded, minimal surfaces of the 3-sphere discrete?
geometry
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Definition of $d (P (x ,y )dx)$

I know it is defined as $ dP \wedge dx $ or explicitly, $$ \frac{\partial P }{\partial y } dy \wedge dx . $$ The question is, could it be $dx \wedge d P $? Or $$ \frac{\partial P }{\partial y } dx \wedge dy ? $$ I know the question must be…
pie
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Evaluating composition of tensor fields $g(X,J(Y))$

Let $M$ be a smooth manifold, $g$ a Riemannian metric and $J$ an almost complex structure on $M$. Since $g$ is a $(0,2)$ tensor field, we get $$g(X,Y) \in C^\infty(M)$$ for all $X,Y \in \mathfrak{X}(M)$. Furthermore, since $J$ is a $(1,1)$ tensor…
TheGeekGreek
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Is $V\otimes V\otimes V^*$ and $V\otimes V^*\otimes V$ the same?

Are $V\otimes V\otimes V^*$ and $V\otimes V^*\otimes V$ the same? I think tensor product is commutative and associative, so I think they are the same thing, then why is it necessary to use different notation $T_2^1$ and $T_{1\;1}^1$ do distinguish…
hxhxhx88
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