# What is Differential geometry: Definition and 409 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.

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1. ### I Rewriting Equation of Motion in terms of Dual Fields (Chern-Simons)

I am reading the following notes: https://arxiv.org/pdf/hep-th/9902115.pdf and am trying to make the connection between equations (22) and (24). Specifically, I do not understand how they were able to get (24) from (22) using the dual field prescription. I guess naively I'm not even sure where...
2. ### Differential geometry of singular spaces

TL;DR Summary: Reference request Hello! Reading the book "Differential geometry of Singular Spaces and Reduction of symmetry" by J. Sniatycki https://www.cambridge.org/core/books/differential-geometry-of-singular-spaces-and-reduction-of-symmetry/7D73498C35A5975594605428DA8F9267 I found that...
3. ### I The Road to Reality - exercise on scalar product

Hi, I'm keep studying The Road to Reality book from R. Penrose. In section 12.4 he asks to give a proof, by use of the chain rule, that the scalar product ##\alpha \cdot \xi=\alpha_1 \xi^1 + \alpha_2 \xi^2 + \dots \alpha_n \xi^n## is consistent with ##df \cdot \xi## in the particular case...

40. ### I Maximally Symmetric 3-Spaces

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?
41. ### A Geodesics of the 2-sphere in terms of the arc length

I'm trying to evaluate the arc length 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 length parameterization of the...
42. ### I Differential Geometry: Comparing Metric Tensors

Is there ever an instance in differential geometry where two different metric tensors describing two completely different spaces manifolds can be used together in one meaningful equation or relation?

45. ### I Geodesics subject to a restriction

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}##...
46. ### Geometry Differential Geometry: Book on its applications?

Hi, I'm already familiar with differential forms and differential geometry ( I used multiple books on differential geometry and I love the dover book that is written by Guggenheimer. Also used one by an Ian Thorpe), and was wondering if anyone knew a good book on it's applications. Preferably...
47. ### I Curve Inside a Sphere: Differentiating Alpha

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.
48. ### A Derive Radial Momentum Eq. in Kerr Geometry

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...
49. ### I Understanding the definition of derivative

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...
50. ### A I need some fun questions with answers in differential geometry ()

I am throwing a bachelor party for my brother, who is currently getting his PhD in Math at columbia, and as you might expect, he is not very much of a party animal. I want to throw him a party he’ll enjoy, so I came up with scavenger hunt in the woods, where every step in the scavenger hunt is a...