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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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How Ricci Flow makes room to find Enistein metrics?

I am studding a lecture note entitled "Topics in Riemanian Geometry" by Jeff. Viaclovsky. See the below phrase in lecture 12: "In order to find Einstein metrics, one would first think of looking at the gradient Ricci flow on the space of Riemannian…
Ramand
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Does the geodesic on a surface $z = f(x,y)$ always trace out a straight line in the $xy$ plane?

Let $z = f(x,y)$ be a surface. Let $(x_0, y_0, z_0)$ and $(x_1, y_1, z_1)$ be two points on that surface. Let $g(t) = \langle x(t), y(t), z(t)\rangle$ be a parameterization of the geodesic curve between the two points. Is the following statement…
Michael Stachowsky
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Differential Geometry: Map Vs Trace

I have started reading a book on the subject and the author writes: "one should carefully distinguish a parametrized curve, which is a map, from its trace, which is a subset of $\mathbb{R}^3$" What does he mean? If we look at $\alpha(t)=cost,sint$…
gbox
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Parameterization of a rhombus

The graph of curve $|x|+|y|=1 $ is a rhombus, How can I obtain a parameterization in a counterclockwise sense in such a way that it can be expressed as the curve $\alpha:I\subset R\to R^2 $,$\alpha (t)=(x(t),y(t))$ ?
Victor Huuanca Sullca
  • 455
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Exponentiation of a vector field

I study the book "Geometry topology and physics" by Nakahara. And there is something I misunderstand at page 191. Here, we compute by using Taylor series, the flow of a vector field $X$ : $$ \sigma^\mu(t,x) = exp(t \frac{d}{ds})…
StarBucK
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Question About Differential Functions Between Manifolds

Let $X$, $Y$ be manifolds, with respective coordinate charts $\{(U_{\alpha}, \varphi_{\alpha})\}_{\alpha\in I}$ and $\{(V_{\beta}, \psi_{\beta})\}_{\beta\in J}$. I want to show that $f:X\to Y$ is differentiable if and only if for every $C^{\infty}$…
roo
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Geodesic in hyperbolic plane

I'm trying to show that $\gamma(t) = (0,t)$ is a geodesic in the hyperbolic plane, that is for $\mathbb{R}^2$ equipped with the metric $g_{11}=g_{22} = \frac{1}{y^2}$, $g_{12}=0$. The way I was trying to do this was by computing the associated…
user291678
  • 565
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On the definition of a surface

When I read a book of differential geometry for undergraduate students. A surface in $\mathbb R^3$ is a subject that "locally looks like" a piece of $\mathbb R^2$. So we can have locally parametrization on the surface. But a smooth curve is defined…
ZERO
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Good and bad tensors (and the metric)

My question is the following : Why is the metric a good tensors that transforms "well". A "good" tensor is a tensor that transforms like this : I take a tensor $T^{\mu \nu}_\rho$ I thus have : $T=T^{\mu \nu}_\rho \partial_\mu \partial_\nu…
StarBucK
  • 689
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Proving vector space is left invariant by definition

Define a smooth vector field $X_S$ on the orthogonal group, $O(n)$, (group of matrices such that $A^{T} = A^{-1}$), where $X_S (A) = SA$, where $S$ belongs to the space of skew symmetric matrices ( $S^{T} = - S$). Why is $X_S$ left invariant? I…
Dragonite
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taylor expansion on the unit sphere

I am looking for a taylor type expansion for functions defined on the unit sphere $S^2$, is the following correct or what should be the right form: $$f(y)-f(x)=\langle \nabla_g f(x), \gamma\rangle \Theta+higher\,\,order\,\, terms$$ where $\nabla_g$…
geometer007
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the existence of a function

How can I prove that if $p$ is a critical point of a real-valued fuction $f$, then there is a function $H:T_{(p)}M\to T_{(p)}M$, such that $H$ is bilinear, simetric and: $\bullet$ $H(X_p, Y_p)=X_p(Yf)=Y_p(Xf)$ for all $X,Y$ vector fields. $\bullet$…
Patricia Suarez Gonzalez
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meridian of a surface of revoltion

I'm trying to show that a meridian of a surface of revolution is a geodesic, except I cannot do it without solving a system of differential equations. And, how can we determine which circles of latitute are geodesics? Thanks
mary
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Linearly independence and Smooth Local Section

Let $\pi:E\rightarrow M$ a smooth vector bundle, $U$ a neighborhood of $p$ in $M$ and $\sigma_i:U\rightarrow E, i=1,\dots n$ smooth local sections of $E$. If $\{ \sigma_1(p), \dots, \sigma_n(p)\}$ are linearly independent (as elements of the vector…
perlman
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curve on the unit sphere

Suppose a curve on the unit sphere intersects every great circle on this sphere,and this curve has length $2\pi$,then prove that this curve must be the union of two halves great circles. I know that a curve on the unit sphere with length less than…
Jack
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