What is Differential geometry: Definition and 420 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. J

    A relatively easy differential geometry question concerning principle curvatures

    Homework Statement Show the principal curvatures on x sin z - y cos z = 0 are +-1/(1 + x^2 + y^2)
  2. G

    Differential geometry: smooth atlas of an ellipsoid

    Homework Statement Consider the ellipsoid L \subsetE3 specified by (x/a)^2 + (y/b)^2 + (z/c)^2=1 (a, b, c \neq 0). Define f: L-S^{2} by f(x, y, z) = (x/a, y/b. z/c). (a) Verify that f is invertible (by finding its inverse). (b) Use the map f, together with a smooth atlas of S^{2}, to...
  3. A

    Book for differential geometry

    HI, am a newbie to differential geometry..Can anyone please suggest me a book suitable for Maths hons student... Before posting read this out... required topics- one parameter family of surfaces, developables associated with a curve : polar and rectifying & osculating developables ,two...
  4. I

    Difficulty of Topology vs Differential Geometry

    So I need to decide by tomorrow, whether I'll be taking topology or diff geo, (along with real analysis and advanced linear algebra). I've sat in on both classes for the first lecture, and I'm still not certain which class would be more difficult. My diff geo class has no exams, and instead...
  5. D

    Tensors, metrics, differential geometry, and all that

    I'm looking to learn general relativity, but I'm having a hard time. Frankly, I can't find any textbooks that I can understand. There seems to be a gap between the maths I did at uni, and the maths of general relativity. I've done vector calculus, differential equations, linear algebra and...
  6. MathematicalPhysicist

    Books on differential geometry on Banach Spaces.

    Can you recommend me of books or preprints that cover reasonabely well this topic? Thanks.
  7. Rasalhague

    Fecko: Differential Geometry 2.3.1

    According to the preceding paragraph, x^i(t) \equiv x^i (\gamma (t)). For now, I have the lowly ambition of trying to understand the notation. I think the xi on the left of the "quoted" equation is a coordinate presentation of a curve (Fecko's curves are functions from an interval to a...
  8. Z

    Applying Algebraic Topology, Geometry to Nonabelian Gauge Theory

    I 've been reading about Homotopy , homology and abstract lie groups and diff.forms and I would like to see those beautiful ideas applied on a Nonabelian Gauge Theory . Any recommendations for a textbook that apply these ideas to gauge theory ? Text books on particle Physics and QFT do not...
  9. WannabeNewton

    Book Recommandation - Differential Geometry

    Hi guys. I finished working through D'Inverno's "Introducing Einstein's Relativity" and Schutz's "A First Course in General Relativity" and some of Carroll's "Spacetime and Geometry" but I don't really feel like I learned most of what is out there. I also feel that before I can tackle Wald I...
  10. D

    How applicable is graduate-level Differential Geometry to GR?

    I'm an undergraduate student who is trying to decide whether to focus on mathematics or physics. I'd like to know how much Differential Geometry is applicable to GR? If I were to take rigorous courses in Differentiable Manifolds and Differential Geometry, will these courses allow a deeper...
  11. T

    Looking for books on group theory and differential geometry

    My university doesn't offer many courses on theoretical physics (I'm studying applied physics), but because I might want to get my masters degree in theoretical physics, I want to read into some of the math and physics. What books would you recommend to a student who has had linear algebra...
  12. R

    Differential geometry acceleration as the sum of two vectors

    Homework Statement a(t)=<1+t^2,4/t,8*(2-t)^(1/2)> Express the acceleration vector a''(1) as the sum of a vector parallel to a'(1) and a vector orthogonal to a'(1) Homework Equations The Attempt at a Solution I took the first two derivatives and calculated a'(t)=<2t, -4t^2, -4/(2-t)^(1/2)>...
  13. M

    Differential Geometry (do Carmo) proof

    Homework Statement I'm going over a proof in Differential Geometry of Curves and Surfaces by do Carmo, and I don't know why the proof can't be shortened to my proof given below. (Proposition 9 on page 130) Proposition. Let f:U-->R be a differentiable function defined on a connected open subset...
  14. K

    Differential Geom: Determining Partial Derivatives of f

    I have been working on determining which partial derivative exists for the surface z=y. i.e. ( partial of f in respect to x, partial of f in respect to y, partial of f in resprct to z). The function f= x^2 -y-z. I think the only ones that exist would be the partial in respect to y and the...
  15. C

    Computational Proofs for Tangent Vectors and Differential Geometry of Surfaces

    Homework Statement Given the following: Some surface M: z=f(x,y) where f(0,0)=fx(0,0)=fy(0,0) and U =-f1U1-fyU2+U3}/Sqrt[1+fx2+fy2] and u1 = U1(0) u2 = U2(0) are vectors tangent to M at the origin 0. We want to prove S(u1)=fxxu1+fxyu2 My problem here is conceptually wadding through this...
  16. N

    Differential Geometry in physics

    My school offers this course at the senior undergraduate/graduate level, and its only offered the semester before General Relativity is offered. Would taking this course really "help out" that much with the mathematics of GR? I am trying to select a few math courses to take that could possibly...
  17. O

    Minimal Surfaces, Differential Geometry, and Partial Differential Equations

    Last night in a lecture my professor explained that some partial differential equations are used to observe events on minimal surface (e.g. membranes). A former advisor, someone that studied differential geometry, gave a brief summary of minimal surfaces but in a diffy G perspective. 1.)...
  18. T

    Differential Geometry: the Osculating Circle

    Homework Statement let f: J --> R^2 be a unit speed curve curve and define it's tangentially equidistant campanion by g(u) = f(t) + r*f'(t) for a fixed r>0. Show that the centre of the osculating circle of f at some u in J is the intersection of the line normal to f'(u) through f(u) and the...
  19. L

    Partial Diff. equations & Differential Geometry?

    I have two classes that I think I can fit in on my last semester at university. I am doing the combined Mathematics & Physics major. I am more interested in physics. However, the two classes I can fit, happen to be mathematics classes. PDE's seem like they are very useful for physics and...
  20. T

    Differential geometry for a physicist?

    Hi, I'm looking to apply for theoretical physics PhDs in the coming year and have been recommended getting a little more mathematics under my belt. I have already done a bit of differential geometry in my G.R. course but I don't think it went into enough depth to be useful outside of what we...
  21. T

    Physics Differential Geometry in Physics

    Hi guys, what are the fields of theoretical physics (if any) -besides General Relativity, String Theory, Quantum Gravity...- where Differential Geometry and Tensorial Calculus currently find strong application? Thanx
  22. P

    Is classical differential geometry still useful?

    Is "classical" differential geometry still useful? As a physics major I have seen in general relativity the power of modern differential geometry such as coordinate-free treatment of manifolds and Riemannian geometry. However, I've also encountered math textbooks devoted to "classical"...
  23. qspeechc

    Any recommendations for a Differential Geometry book?

    Hi everyone. I am a senior undergrad math major and I'm looking for a Differential Geometry book to self-study. I have studied most/all of the other undergrad topics: algebra; real and complex analysis; point-set topology; etc. Any recommendations? Thanks
  24. S

    Differential Geometry: Coordinate Patches

    Sorry i wasnt able to get help in the homework department. figured id try here. Homework Statement For a coordinate patch x: U--->\Re^{3}show thatu^{1}is arc length on the u^{1} curves iff g_{11} \equiv 1 The Attempt at a Solution So i know arc legth of a curve \alpha (t) =...
  25. S

    Differential geometry: coordinate patches

    Homework Statement For a coordinate patch x: U--->\Re^{3}show thatu^{1}is arc length on the u^{1} curves iff g_{11} \equiv 1 The Attempt at a Solution So i know arc legth of a curve \alpha (t) = \frac{ds}{dt} = \sum g_{ij} \frac {d\alpha^{i}}{dt} \frac {d\alpha^{j}}{dt} (well that's actually...
  26. O

    Differential geometry or number theory (which to take)

    hi, I'm entering my 3rd year of PMAT degree and need to make a choice between differential geometry and number theory. These are both undergrad courses. I am trying to decide which would be more interesting/useful to take. I am planning on going into grad school, so it would be nice to choose a...
  27. Fredrik

    How can we use differential geometry to improve our understanding of SR?

    When I say "SR" in this post, I mean the set of classical and quantum theories of particles and fields in Minkowski spacetime. I'm trying to come up with a list of topics in SR that can be dealt with in a better way when we have defined Minkowski spacetime as a manifold instead of as a vector...
  28. B

    Differential Geometry: Unit Normal Field

    Let M be the surface defined by z=x2+3xy-5y2. Find a unit normal vector field U defined on a neighborhood of p on M. First, I reparameterized the equation for the surface to get x(u,v)=(u,v,u2+3xy-5y2). Next I found two tangent vectors xu(u,v)=(1,0,2u+3v) and xv=(0,1,3u-10v). The next step is...
  29. F

    Introductory Book on Differential Geometry

    Hey, I was wondering if anyone could recommend an introductory book for differential geometry. I am studying general relativity and need some help with this topic. Thanks.
  30. P

    Differential geometry question

    Homework Statement A function F of n real variables is called homogeneous of degree r if it satisfies F(tx_1, tx_2, ..., tx_n) = (t^r)F(x_1,x_2,...,x_n) By differentiation with respect to t, show that a function F is an eigenfunction of the operator: x^1 ∂/∂x_1 + ... + x^n ∂/∂x_n...
  31. J

    Help with differential geometry: Hilbert's stress energy tensor

    When varying the Hilbert action, we define the stress-energy tensor as: T_{\mu\nu}:= \frac{-2}{\sqrt{-g}}\frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{matter})}{\delta g^{\mu\nu}} = -2 \frac{\delta \mathcal{L}_\mathrm{matter}}{\delta g^{\mu\nu}} + g_{\mu\nu} \mathcal{L}_\mathrm{matter} I am...
  32. Z

    Bridge between complex analysis and differential geometry

    I am not a mathematician but I have noticed how strangley similar the treatments of curvature and residues are when you compare the residues of residue calculus and the curviture of the gauss bonet forumlation of surfaces. Is there some generalization of things that contains both of these...
  33. S

    Differential Geometry: Learn Its Importance for Physics

    How important is to learn differential geometry to do Physics?
  34. Y

    Differential geometry reconstructed: a unified systematic framework

    The first few sections of this book gave me a good sense of the solid foundation that the author is building in this book. While it may be review for many, this guy writes so well, it's a pleasure to read...
  35. I

    Patches and Surfaces (Differential Geometry)

    I'm completely confused with patches, which were introduced to us very briefly (we were just given pictures in class). I am using the textbook Elementary Differential Geometry by O'Neill which I can't read for the life of me. I'm here with a simple question and a somewhat harder one...
  36. B

    A differential geometry question.

    Fix a number L > 2. Consider all smooth plane curves r of length L that connect (−1, 0) and (1, 0) and are contained in the upper half-plane. Note that and the segment [−1, 1] of the X-axis together bound a plane domain D. Find r such that D has the maximal possible area.
  37. R

    Differential Geometry and Quantum Mechanics

    After finding and reading Geroch's notes on Quantum Mechanics formulated within Differential Geometry, I was wondering if there are other books that treat Quantum Mechanics in a similar fashion, focusing upon the geometrical aspects of Quantum Mechanics in order to formulate it.
  38. S

    Differential Geometry Problems (2)

    Homework Statement 1. 2. Homework Equations Frenet Formulas, definitions of curvature, torsion and generalized helix The Attempt at a Solution for 1) I think I got part A down - I had α = λT + µN + νT, took the derivatives and plugged in the Frenet formulas to get: λ′ − µκ...
  39. C

    Compute Frenet apparatus (differential geometry)

    Homework Statement Find the Frenet apparatus for the curve \alpha (t) = (at, bt^2, ct^3), where abc \neq 0. Homework Equations The Frenet equations The Attempt at a Solution The derivative of the curve is the expression for the tangent vector. The second derivative (the first...
  40. P

    Differential geometry - strange formula

    heeello friends!;] i have book "wstęp do współczesnej geometrii różniczkowej" written by Konstanty Radziszewski, this title in english mean something like... "basics of modern differential geometry", and here are many formulas which look similar to one another, but I never know what it means and...
  41. B

    A differential geometry question

    Show that the knowledge of the vector function n = n(s) (normal vector) of a unit-speed curve , with non-zero torsion everywhere, determines the curvature and the torsion don't have any clues about what i am supposed to prove!~
  42. K

    [Differential Geometry] Simple relations between Killing vectors and curvature

    First of all, hello :) I'd like to request some aid concerning a problem that is really getting to me. I know it should be simple but I'm not getting the right results. Homework Statement Given that V^{\mu} is a Killing vector, prove that: V^{\mu;\lambda}_{;\lambda} +...
  43. D

    Differential Geometry - useful in engineering?

    Dear fellow Mathematicians and Physicists, As the fall term closes and spring term starting next year, I am deciding on which math class to take. Simply put, I'm a math major (possibly engineering if there are enough slots in my schedule), more applied, seeking to apply math concepts to...
  44. T

    Matrix Formalism of Differential Geometry

    I would like to explore writing differential geometry in matrix format and was wondering if any of the experts here knows a good resource for that? I have tried Google and can't find anything definitive. Thanks in advance!
  45. S

    Differential Geometry Question

    Homework Statement Consider the following parametrization of a Torus: \sigma(u,v)=((R+r\cos u)\cos v, (R+r\cos u)\sin v, r\sin u) R>r,\quad (u,v)\in [0,2\pi)^2 1. Compute the Gauss map at a given point. 2. What are the eigenvalues of that map in the base...
  46. T

    Best foreign language to study for differential geometry?

    I want to be able to read bold research that hasn't yet been translated into English. There are so many wide-open problems out there in this field...
  47. L

    Differential geometry and hamiltonian dynamics

    Hello everybody! I'm currently attending lectures on Hamiltonian dynamics from a very mathematical viewpoint and I'm having trouble understanding two facts: 1. An inner product defined in every tangent space and a symplectic form both establish a natural isomorphism between tangent and...
  48. T

    For professionals - differential geometry

    Hello guys, I keep hearing that Euclidean parallel postulate was broken through differential geometry, can someone please explain how that happens, and in what sense? I do understand differential geometry notations and tensors, so explanation with them is OK. Thank you :)
  49. S

    Help with differential geometry

    I am going to be entering the university of texas graduate physics department in August. I am currently signed up for the class "Topics in Geometry and Quantum Physics" (http://www.ma.utexas.edu/users/dafr/M392C/index.html) and am pretty worried about the summer reading. I am having a hard time...
  50. S

    Differential Geometry Theorem on Surfaces

    Homework Statement I am having difficulty understanding the proof of the following theorem from Differential Geometry Theorem S\subset \mathbb{R}^3 and assume \forall p\in S \exists p\in V\subset\mathbb{R}^3 V open such that f:V\rightarrow\mathbb{R}^3 is C^1 V\cap S=f^{-1}(0)...
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