differential geometry Definition and Topics - 167 Discussions
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.
Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques endemic to this field.
I am trying to see if Vector fields(I am thinking of electric and magnetic fields) without sources(divergence less) can be represented by a pair of functions f and g such that the level surfaces of the functions represent flux lines. I am trying to solve this problem in ## R^3 ## with a...
I was just browsing through the textbooks forum a few days ago when I came across a post on differential geometry books.
Among the others these two books by the same author seem to be the most widely recommended:
Elementary Differential Geometry (Barret O' Neill)
Semi-Riemannian Geometry with...
In my notes on general relativity, hypersurfaces are defined as in the image. What confuses me is that if f=constant, surely the partial differential is going to be zero? I'm not sure if I'm missing something, but surely the function cant be equal to a constant and its partial differential be...
I know the osculating plane is normal to the binormal vector ##B(t)=(a,b,c)##. And since the point on which I am supposed to find the osculating plane is not given, I'm trying to find the osculating plane at an arbitrary point ##P(x_0,y_0,z_0)##. So, if ##R(x,y,z)## is a point on the plane, the...
Greetings. I still struggle a little with the mathematics involved in the description of gauge theories in terms of fiber bundles, so please pardon and correct me if you find conceptual errors anywhere in this question. I would like to understand the connection (when it exists) between the...
i'm trying to find what sort of 2-d geometry this system is in, i've been given the line element
𝑑𝑠2=−sin𝜃cos𝜃sin𝜙cos𝜙[𝑑𝜃2+𝑑𝜙2]+(sin2𝜃sin2𝜙+cos2𝜃cos2𝜙)𝑑𝜃𝑑𝜙
where
0≤𝜙<2𝜋
and
0≤𝜃<𝜋/2
Im just not sure where to start. Ive tried converting the coordinates to cartesian to see if it yields a...
Let ## \mathcal{S} ## be a family of probability distributions ## \mathcal{P} ## of random variable ## \beta ## which is smoothly parametrized by a finite number of real parameters, i.e.,
## \mathcal{S}=\left\{\mathcal{P}_{\theta}=w(\beta;\theta);\theta \in \mathbb{R}^{n}...
I read in the book Gravitation by Wheeler that "Any tensor can be completely symmetrized or antisymmetrized with an appropriate linear combination of itself and it's transpose (see page 83; also this is an exercise on page 86 Exercise 3.12).
And in Topology, Geometry and Physics by Michio...
Could you provide recommendations for a good modern introductory textbook on differential geometry, geared towards physicists. I know physicists and mathematicians do mathematics differently and I would like to see how it is done by a physicists standard. I have heard Chris Ishams “Modern Diff...
There are a few different textbooks out there on differential geometry geared towards physics applications and also theoretical physics books which use a geometric approach. Yet they use different approaches sometimes. For example kip thrones book “modern classical physics” uses a tensor...
Definition: Let f be a differentiable real-valued function on ##\mathbf{R}^3##, and let ##\mathbf{v}_P## be a tangent vector to it. Then the following number is the derivative of a function w.r.t. the tangent vector
$$\mathbf{v}_p[\mathit{f}]=\frac{d}{dt} \big( \mathit{f}(\mathbf{P}+ t...
Why does the constraint:
$$R_{ijkl}=K(g_{ik} g_{jl} - g_{il}g_{jk})$$
Imply that the resulting space is maximally symmetric? The GR book I'm using takes this relation more or less as a definition, what is the idea behind here?
I'm trying to evaluate the arc lenght between two points on a 2-sphere.
The geodesic equation of a 2-sphere is:
$$\cot(\theta)=\sqrt{\frac{1-K^2}{K^2}}\cdot \sin(\phi-\phi_{0})$$
According to this article:
http://vixra.org/pdf/1404.0016v1.pdf
the arc lenght parameterization of the...
I'm a bit confused about the notation used in the exercise statement, but if I'm not misunderstanding we have
$$\begin{align*}(\psi^+_1)^{-1}:\begin{array}{rcl}
\{\lambda^1,\lambda^2\in [a,b]\mid (\lambda^1)^2+(\lambda^2)^2<1\}&\longrightarrow& \{\pm x_1>0\}\subset \mathbb{S}^2\\...
Hi, I'm trying to solve a differential geometry problem, and maybe someone can give me a hand, at least with the set up of it.
There is a particle in a 3-dimensional manifold, and the problem is to find the trajectory with the smallest distance for a time interval ##\Delta t=t_{1}-t_{0}##...
Honestly I don't know where to begin. I started differentiating alpha trying to show that its absolute value is constant, but the equation got complicated and didn't seem right.
I am trying to derive the radial momentum equation in the equatorial Kerr geometry obtained from the equation $$ (P+\rho)u^\nu u^r_{;\nu}+(g^{r\nu}+u^ru^\nu)P_{,r}=0 \qquad $$. Expressing the first term in the equation as $$ (P+\rho)u^\nu u^r_{;\nu}=(P+\rho)u^r u^r_{;r} $$ I obtained the...
As far as I understand, when we want to differentiate a vector field along the direction of another vector field, we need to define either further structure affine connection, or Lie derivative through flow. However, I don't understand why they are needed. If we want to differentiate ##Y## in...
Let us consider Ashtekar's definition of asymptotic flatness at null infinity:
I want to see how to construct the so-called Bondi coordinates ##(u,r,x^A)## in a neighborhood of ##\mathcal{I}^+## out of this definition.
In fact, a distinct approach to asymptotic flatness already starts with...
For a function ##f: \mathbb{R}^n \to \mathbb{R}##, the following proposition holds:
$$
df = \sum^n \frac{\partial f}{\partial x_i} dx_i
$$
If I understand right, in the theory of manifold ##(df)_p## is interpreted as a cotangent vector, and ##(dx_i)_p## is the basis in the cotangent space at...
Since in 2D the riemman curvature tensor has only one independent component, ## R = R_{ab} g^{ab} ## can be reversed to get the riemmann curvature tensor.
Write
## R_{ab} = R g_{ab} ##
Now
## R g_{ab} = R_{acbd} g^{cd}##
Rewrite this as
## R_{acbd} = Rg_{ab} g_{cd} ##
My issue is I'm not...
Hi everyone! I have a problem with one thing.
Let's consider the Lorentz group and the vicinity of the unit matrix. For each ##\hat{L}##
from such vicinity one can prove that there exists only one matrix ##\hat{\epsilon}## such that ##\hat{L}=exp[\hat{\epsilon}]##. If we take ##\epsilon^{μν}##...
Homework Statement
This problem is from V.I Arnold's book Mathematics of Classical Mechanics.
Q) Show that every differential 1-form on line is differential of some function
Homework Equations
The differential of any function is
$$df_{x}(\psi): TM_{x} \rightarrow R$$
The Attempt at a Solution...
Hi,
I'm just starting to read Wald and I find the notion of the commutator hard to grasp. Is it a computation device or does it have an intuitive geometric meaning? Can anyone give me an example of two non-commutative vector fields?
Thanks!
Hello,
In the sources I have looked into (textbooks and articles on differential geometry), I have not found any abstract definition of the electromagnetic fields. It seems that at most the electric field is defined as
$$\bf{E}(t,\bf{x}) = \frac{1}{4\pi \epsilon_0} \int \rho(t,\bf{x}')...
This is problem 4.7.11 of O'Neill's *Elementary Differential Geometry*, second edition. The hint says to use the Hausdorff axiom ("Distinct points have distinct neighborhoods") and the results of fact that a finite intersection of neighborhoods of p is again a neighborhood of p.
Here is my...
Hi,
consider an "half-cone" represented in Euclidean space ##R^3## in cartesian coordinates ##(x,y,z)## by: $$(x,y,\sqrt {x^2+y^2})$$
It does exist an homeomorphism with ##R^2## through, for instance, the projection ##p## of the half-cone on the ##R^2## plane. You can use ##p^{-1}## to get a...
Hello
I have a question if it possible,
Let X a tangantial vector field of a riemannian manifolds M, and f a smooth function define on M.
Is it true that X(exp-f)=-exp(-f).X(f)
And div( exp(-f).X)=exp(-f)〈gradf, X〉+exp(-f)div(X)?
Thank you
Hi,
Let ##M^3## be a 3-manifold embedded in ##\mathbb R^3## and consider a 2-plane field ( i.e. a Contact Structure) assigned at each tangent space ##T_p##. I am trying to understand obstructions to defining the plane field as a 1-form ( Whose kernel is the plane field/ Contact Structure) Given...