What is Differential geometry: Definition and 419 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. P

    Checking derivation of the curvature tensor

    Homework Statement I am trying to derive the curvature tensor by finding the commutator of two covariant derivatives. I think I've got it, but my head is spinning with Nablas and indices. Would anyone be willing to check my work? Thanks Homework Equations I am trying to derive the curvature...
  2. AXidenT

    Functional Analysis or Differential Geometry?

    I'm in my last semester of my undergraduate majoring in mathematics (focusing on mathematical physics I guess - I'm one subject short of having a physics major) and am wondering, largely from a physics perspective if it would be better to do a functional analysis course or a differential...
  3. U

    Total derivative involving rigid body motion of a surface

    This stems from considering rigid body transformations, but is a general question about total derivatives. Something is probably missing in my understanding here. I had posted this to math.stackexchange, but did not receive any answers and someone suggested this forum might be more suitable. A...
  4. D

    Local parameterizations and coordinate charts

    I have recently had a lengthy discussion on this forum about coordinate charts which has started to clear up some issues in my understanding of manifolds. I have since been reading a few sets of notes (in particular referring to John Lee's "Introduction to Smooth Manifolds") and several of them...
  5. D

    General relativity and curvilinear coordinates

    I have just been asked why we use curvilinear coordinate systems in general relativity. I replied that, from a heuristic point of view, space and time are relative, such that the way in which you measure them is dependent on the reference frame that you observe them in. This implies that...
  6. R

    Help with Proving Ruled Surface Equation

    Hello, I am studying for an analytic geometry final but I am totally lost for this problem... We didn't even cover this topic in class (my prof didn't show up for class for two weeks) and I have no clue on how to do it. If anyone could help I would appreciate it. Question: Prove that the...
  7. D

    Differential map between tangent spaces

    I've been struggling since starting to study differential geometry to justify the definition of a one-form as a differential of a function and how this is equal to a tangent vector acting on this function, i.e. given f:M\rightarrow\mathbb{R} we can define the differential map...
  8. D

    Understanding the Tensor Product of Two One-Forms in Differential Geometry

    I'm relatively new to differential geometry and would like to check that this is the correct definition for the tensor product of (for simplicity) two one-forms \alpha,\;\beta\;\;\in V^{\ast} : (\alpha\otimes\beta)(\mathbf{v},\mathbf{w})=\alpha (\mathbf{v})\beta (\mathbf{w}) where...
  9. X

    Research in Differential Geometry

    I am currently looking at grad schools, and I am wondering if anyone knew who are the leading researchers in differential geometry. I know that question is a little vague considering how diverse differential geometry is, but I was hoping that something could direct me in the right direction...
  10. D

    What is the interpretation of dx in calculus?

    Apologies if this isn't quite the right forum to post this in, but I was unsure between this and the calculus forum. Something that has always bothered me since first learning calculus is how to interpret dx, essentially, what does it "mean"? I understand that it doesn't make sense to consider...
  11. D

    Integrating Forms on Manifolds: Understanding the Concept and Techniques

    In all the notes that I've found on differential geometry, when they introduce integration on manifolds it is always done with top forms with little or no explanation as to why (or any intuition). From what I've manage to gleam from it, one has to use top forms to unambiguously define...
  12. D

    Why are vectors defined in terms of curves on manifolds

    What is the motivation for defining vectors in terms of equivalence classes of curves? Is it just that the definition is coordinate independent and that the differential operators arising from such a definition satisfy the axioms of a vector space and thus are suitable candidates for forming...
  13. D

    Tangent vectors as directional derivatives

    I have a few conceptual questions that I'd like to clear up if possible. The first is about directional derivatives in general. If one has a function f defined in some region and one wishes to know the rate of change of that function (i.e. its derivative) along a particular direction in that...
  14. D

    Attempting to understand diffeomorphisms

    I am relatively new to the concept of differential geometry and my approach is from a physics background (hoping to understand general relativity at a deeper level). I have read up on the notion of diffeomorphisms and I'm a little unsure on some of the concepts. Suppose that one has a...
  15. D

    Diffeomorphisms and active transformations

    I've been reading Sean Carroll's notes on General Relativity, http://arxiv.org/pdf/gr-qc/9712019.pdf . I've got to chapter 5 (page 133) and am reading the section on diffeomorphisms in which Sean relates diffeomorphisms to active transformations. When he says this does he mean that one defines a...
  16. stevendaryl

    How Do Spinors Fit in With Differential Geometry

    When I studied General Relativity using Misner, Thorne and Wheeler's "Gravitation", it was eye-opening to me to learn the geometric meanings of vectors, tensors, etc. The way such objects were taught in introductory physics classes were heavily dependent on coordinates: "A vector is a collection...
  17. PWiz

    Where do I start learning differential geometry?

    I've recently finished tackling differential equations. I want to start learning general relativity, but from what I've read, I need to have a firm footing in differential geometry first. So where do I start learning DG? I really don't want to do a half-hearted job in an attempt to quickly jump...
  18. hideelo

    Relationship between thermodynamics and differential geometry

    I am taking thermodynamics this semester as well as a course in differential geometry of surfaces, and I am seeing a lot of overlap. For example, I can create a "state space" isomorphic to R3 of TxPxV I can then define a surface on this space of PV=NkT I can define quasi static state equations...
  19. D

    A question on defining vectors as equivalence classes

    I understand that a tangent vector, tangent to some point p on some n-dimensional manifold \mathcal{M} can defined in terms of an equivalence class of curves [\gamma] (where the curves are defined as \gamma: (a,b)\rightarrow U\subset\mathcal{M}, passing through said point, such that \gamma (0)=...
  20. D

    Understanding the notion of a tangent bundle

    I've been reading up on the definition of a tangent bundle, partially with an aim of gaining a deeper understanding of the formulation of Lagrangian mechanics, and there are a few things that I'm a little unclear about. From what I've read the tangent bundle is defined as the disjoint union of...
  21. D

    Questions about tangent spaces & the tangent bundle

    This is a slightly physics oriented question, so apologies for that. Basically, having started studying differential geometry it has started to become a little clearer to me why one can consider the Lagrangian as a function of position and velocity, but I don't feel I'm quite there yet. My...
  22. T

    Any value to Spivak's Differential Geometry set?

    I have the hardback 5 volume set of Spivak's A Comprehensive Introduction to Differential Geometry that is in pretty good shape. Is there any value to that set? I tried looking it up, but I don't really see many people selling whole sets, so I can't tell... Thanks.
  23. DrPapper

    Classical Fundamental Principles of Classical Mechanics - Kai S. Lam

    Hello all, I'm currently taking an upper undergraduate two part Mechanics course using the above mentioned book by its author. He's a great professor and I was wondering if anyone else has checked out this book? It's very math heavy and I'm struggling with some of the language since I haven't...
  24. D

    Differentiability of a function on a manifold

    I am currently working through Nakahara's book, "Geometry, Topology and Physics", and have reached the stage at looking at calculus on manifolds. In the book he states that "The differentiability of a function f:M\rightarrow N is independent of the coordinate chart that we use". He shows this is...
  25. H

    Test Tracks: Banked Parabolic Curve Benefits

    Why test tracks are banked parabolic why not elliptical or circular or hyperbolic? what are advantages of parabolic curve over another curves?
  26. P

    Differential Geometry vs. General Relativity

    Hello, This spring, I will have the opportunity to do a one-on-one independent study in math or physics. I've narrowed down my choices to differential geometry and general relativity. I'm thinking about the future here- will studying general relativity this spring better prepare me for...
  27. E

    Christoffel symbols in differential geometry

    Homework Statement I'm having trouble figuring out how to use Christoffel symbols. Apart from the first three terms here, I can't understand what's going on between line 3 and 4. What formulas/definitions are being used? How do you find the product of two chirstoffel symbols? Where are all the...
  28. V

    One-forms in differentiable manifolds and differentials in calculus

    Suppose that we have this metric and want to find null paths: ds^2=-dt^2+dx^2 We can easily treat dt and dx "like" differentials in calculus and obtain for $$ds=0$$ dx=\pm dt \to x=\pm t Now switch to the more abstract and rigorous one-forms in differentiable manifolds. Here \mathrm{d}t (v)...
  29. V

    Find Null Paths in Differentiable Manifolds Using One-Forms

    Suppose that we have this metric and want to find null paths: ds^2=-dt^2+dx^2 We can easily treat dt and dx "like" differentials in calculus and obtain for $$ds=0$$ dx=\pm dt \to x=\pm t Now switch to the more abstract and rigorous one-forms in differentiable manifolds. Here \mathrm{d}t (v)...
  30. JonnyMaddox

    What is the Geometric Interpretation of Principal Bundles with Lie Group Fibres?

    I'm trying to learn how to think about principal bundles where the fibre is a lie group with local trivialization ϕ^{-1}_i:π(U_i)→U_i×G . For example ϕ^{-1}_i:π(S^2)→S^2×U(1) (if that makes sense) . But I don't know how to think of this (and other products with lie groups like that)...
  31. P

    Antisymmetrization in Wedge Product: Exploring $$\alpha \Lambda \beta$$

    If I want to take the wedge product of $$\alpha = a_i\theta^i $$ and $$\beta = b_j\theta^j$$ I get after applying antisymmetrization,$$ \alpha \Lambda \beta = \frac{1}{2}(a_ib_j - a_jb_i)\theta^i\theta^j$$ My question is it seems to me that antisymmetrization technique doesn't apply to the...
  32. mef51

    [Differential Geometry] Matrix of Differential Equations in SO(3)

    Homework Statement Suppose that ##s \to A(s) \subset \mathbb{M}_{33}(\mathbb{R})## is smooth and that ##A(s)## is antisymmetric for all ##s##. If ##Q_0 \in SO(3)##, show that the unique solution (which you may assume exists) to $$\dot{Q}(s) = A(s)Q(s), \quad Q(0) = Q_0$$ satisfies ##Q(s) \in...
  33. D

    Intro to differential geometry with worked examples

    Hi. I am looking for the most basic intro to differential geometry with plenty of worked examples. I want it to cover the following - differential forms , pull-backs , manifolds , tensors , metrics , Lie derivatives and groups and killing vectors. Problems with solutions would also be good as I...
  34. D

    Different types of differential geometry?

    I am planning on taking a course in differential geometry. I have looked at the notes and they cover - differential forms , pull-backs , tangent vectors , manifolds , Stokes' theorem , tensors , metrics , Lie derivatives and groups and killing vectors. I have a book called Elementary...
  35. M

    Differential geometry of surfaces in affine spaces

    I'm looking for a book or two that details affine spaces and transformations, then differential geometry of surfaces in affine spaces, starting at a level suitable for a year 1-2 undergraduate. In particular, I'd like to understand a few properties (e.g. what's the gradient and curvature at a...
  36. Q

    Differential Geometry book on 3D Euclidn space - worth reading?

    I bought a book (Differential Geometry by Kreyszig) based on really good reviews because I'm planning to learn general relativity later. I guess I didn't pay enough attention to the description because apparently it's completely focused on "three-dimensional Euclidean space." Will this book...
  37. S

    What is the difference b/w analytic, algebraic, differential geometry?

    Also, is there a special term for the geometry typically taught in high school? And how does topology fit in here? It seems that topology is a form of geometry as well.
  38. P

    Differential geometry Frenet

    even γ: I-> R ^ 2 curve parameterized as to arc length (single speed) with curvature k (s)> 0 and torsion τ(s)> 0. I want to write the γ(s) as a combination of n(s), t(s), b(s). these are the types of Frenet. the only thing i know is that the types of Frenet are t(s)=γ'(s) ...
  39. P

    Exploring Tangent Planes and Vertical Curvature in Differential Geometry

    Consider the surface S defined as the graph of a function z = 2x ^ 2 - y ^ 2 i) find a basis of the tangent plane Tp surface S at the point p = (-1,2, -2) ii) find a non-zero vector w in Tp with the property that the vertical curvature at point p in the direction of vector w is zero for...
  40. P

    Is t(s) perpendicular to the radius of the surface sphere at point γ(s)?

    whether γ=γ(s):I->R^3 curve parameterized as to arc length (single speed). Assume that γ is the surface sphere centered on the origin (0,0). Prove that the vector t(s) is perpendicular to the radius of the sphere at point γ(s), for each s. i know that t(s)=γ΄(s) but i don't know how to...
  41. M

    Good differential geometry textbooks/books for self study?

    Any good textbooks on differential geometry for self study? I'm not the best at reading comprehension. I've already studied Calculus, differential equations, and linear algebra.
  42. D

    Maths required to start differential geometry

    I have a Physics background and have done the relevant maths ie. calculus , linear algebra , vector calculus and differential equations. Do i need any "extra maths" before starting a course in differential geometry ? Any recommendations for a book on the subject that would suit a Physicist ? Thanks
  43. M

    What books or aid can I use to learn differential geometry

    I am very curious with what differential geometry is. Can you send me links, books, and etc? I want to learn it. Thank you in advance
  44. L

    Looking for Differential Geometry books

    I am Looking for some books about Differential Geometry,and I just begin to learn Differential Geometry,so who can introduce some books about Differential Geometry that suitable for Beginners to me,and tell me where can download the PDF document of the book. thanks
  45. B

    Differential Geometry for General Relativity and Yang-Mills Theories

    I have been teaching myself QFT and General Relativity. The mathematics of those fields is daunting, and I find that what I have come across is very difficult to master. Of course it will take work, but can someone recommend a good text for self-leaning differential geometry with application...
  46. shounakbhatta

    General relativity without Differential geometry

    Hello, I am learning General Relativity through some books like 'Gravity' by Hartle and through some other textbooks. All those books, do not speak of general relativity from the context of differential geometry. I have a fair amount of knowledge of calculus as well as set theory. My...
  47. Z

    Acceleration in arbitrary trajectory (differential geometry)

    Homework Statement Show that for any trajectory r(t) the acceleration can be written as: \mathbf{a}(t)=\frac{dv}{dt}\hat{T}(t)+\frac{v^2}{\rho}\hat{N}(t) where v is the speed, T is a unit vector tangential to r and N is a unit vector perpendicular to T, at time t. rho is the radius of...
  48. J

    Flow Chart of material to learn differential geometry

    I am a 3rd year mechanical engineering student at LSU, but my true interest lies in theoretical physics and mathematics (specifically general relativity and differential geometry). I've taken calculus 1,2,3, linear algebra, ordinary differential equations, number theory, discrete math, and...
  49. Barioth

    MHB Do carmo Differential Geometry Solution?

    Hi, I was wondering if someone know of a solution book for this book, I'm studying for final, having lot of problem with this class... (not my type of math I guess!) Thanks
  50. D

    Is analysis necessary to know topology and differential geometry?

    I'm a physics major interested in taking some upper level math classes such as topology, differential geometry, and group theory but these classes are only taught in the math department and are heavy on the proofs. Analysis are recommended and preferred prerequisites but are apparently not...
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