What is Differential geometry: Definition and 418 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.

View More On Wikipedia.org
Recent contents Top users
  1. Superposed_Cat

    Where to learn differential geometry online?

    Hi all, I was wondering where I could learn differential geometry online. Preferably via videos. If anyone could post any links to free sites it would be much appreciated. Thanks in advance.
  2. L

    Curves on surfaces (differential geometry)

    A few topics we are covering in class are: Gauss map, Gauss curvature, normal curvature, shape operator, principal curvature. I am having difficulty understanding the concepts of curves on surfaces. For example, this problem: Define the map ##\pi : (\mathbb{R}^3-\{(0,0,0)\})\to S^2## by...
  3. S

    Literature on differential geometry, suggestions?

    I am reading Spivak, Calculus on manifolds, and I have a basic working knowledge of topology through Mendelson, "Introduction to Topology", I want to learn more about differential geometry, especially co variant derivatives, levi-civita connections, Ricci and Rieman curvature tensors. I know...
  4. STEMucator

    Differential geometry, what book is good for a first timer?

    I've been wanting to see what this topic is all about for awhile now. I see the word "Manifold" and other terminology floating around on the forum. It got me really curious. I wonder what the prerequisites are to reading a book like this? Hypothetically I have the prerequisites, what would...
  5. C

    Good treatments on the differential geometry on surfaces.

    Hi! I'm trying to read up on the subject of hypersurfaces related to GR; First and second fundamental form, Theorema Egregium etc.. Does anyone know any good treatments? (Books or notes)
  6. O

    Differential Geometry - Finding Flat Coordinates

    Homework Statement Hello, I posted a similar question in the physics section but no one was able to help, I am first going to include a link to the older problem where I was attempting to find the ,(Finding the local flat space of the Poincare half disk metric), and explain what is different...
  7. P

    What exactly is differential geometry?

    Does it still have a sense of Euclid-style geometry-are there still cubes and spheres, so to speak? Is it mostly about 1D curves/2D surfaces, or does it consider higher dimensions? Are the surfaces which the field concerns mostly graphs of several variables, e.g. ## x^3+y^3+z^3=1 ##, or are they...
  8. Greg Bernhardt

    Geometry Elementary Differential Geometry by Barrett O'Neill

    Author: Barrett O'Neill Title: Elementary Differential Geometry Amazon Link: https://www.amazon.com/dp/0120887355/?tag=pfamazon01-20 Prerequisities: Contents:
  9. S

    Spivak's Differential Geometry I

    Is it necessary to finish Spivak's little book to move on to Spivak's Differential Geometry I, or is the material on differential forms and integration on manifolds in Chapter's 4 and 5 of Spivak's little book covered in Differential Geometry I?
  10. shounakbhatta

    Basic understanding of differential geometry

    Hello, I am new very new in this subject. I have a curiosity in understanding diff.geometry. I have some questions (which might sound elementary) to ask: (1) Is diff.geometry a subject related to the study of surface, curvatures, manifolds? (2) How it is different from Euclidean geometry...
  11. shounakbhatta

    [Differential geometry] Book suggestion required

    Hello, I am a beginner. I am self taught in differential calculus. Can you please suggest me any book, as a beginner, to have a very basic idea and overview on Differential Calculus. Any free e-book? Kindly suggest.
  12. I

    Differential Geometry: angle between a line to a curve and a vector

    Homework Statement Let α(t) be a regular, parametrized curve in the xy plane viewed as a subset of ℝ^3. Let p be a fixed point not on the curve. Let u be a fixed vector. Let θ(t) be the angle that α(t)-p makes with the direction u. Prove that: θ'(t)=||α'(t) X (α(t)-p)||/(||(α(t)-p)||)^2...
  13. J

    Question about differential geometry

    Hi, I read in Padmanabhan's book that \nabla_a J^a=0 implies that there exists an antisymetric tensor P such that J^a= \nabla_b P^{ba}. What's the name of the theorem? Any reference? Thanks
  14. I

    A Basic Differential Geometry Question

    Suppose x(t) is a curve in ℝ^2 satisfying x*x'=0 where * is the dot product. Show that x(t) is a circle. The hint says find the derivative of ||x(t)||^2 which is zero and doesn't tell me much. I was hoping for x*x= r, r a constant.
  15. Telemachus

    Is Lang's Book on Differential Geometry Suitable for Beginners?

    Hi there. I want to learn some differential geometry on my own, when I find some time. My intention is to learn the maths, so then I can get some insight, and go more deeply on the foundations of mechanics. I need to start on the basics. I had some notions on topology when I did my analysis II...
  16. P

    How to understand differential geometry

    the differential geometry is so abstract to understand. All are terms and theorem. How to understand it? can someone give me some method and guidance to learn it. HELP!
  17. micromass

    Geometry Fundamental of Differential Geometry by Lang

    Author: Serge Lang Title: Fundamentals of Differential Geometry Amazon Link: https://www.amazon.com/dp/038798593X/?tag=pfamazon01-20 Prerequisities: Grad Analysis, Differential Geometry Level: Grad Table of Contents: Foreword Acknowledgments General Differential Theory Differential...
  18. micromass

    Geometry Applied Differential Geometry by Burke

    Author: William Burke Title: Applied Differential Geometry Amazon Link: https://www.amazon.com/dp/0521269296/?tag=pfamazon01-20 Prerequisities: Level: Undergrad Table of Contents: Preface Glossary of notation Introduction Tensor in linear spaces Linear and affine spaces...
  19. micromass

    Geometry A Comprehensive Introduction to Differential Geometry series by Spivak

    Author: Michael Spivak Title: A Comprehensive Introduction to Differential Geometry Amazon Link: https://www.amazon.com/dp/0914098705/?tag=pfamazon01-20 https://www.amazon.com/dp/0914098713/?tag=pfamazon01-20 https://www.amazon.com/dp/0914098721/?tag=pfamazon01-20...
  20. micromass

    Geometry Modern Differential Geometry for Physicists by Isham

    Author: C.J. Isham Title: Modern Differential Geometry for Physicists Amazon Link: https://www.amazon.com/dp/9810235623/?tag=pfamazon01-20 Table of Contents: An Introduction to Topology Preliminary Remarks Remarks on differential geometry Remarks on topology Metric Spaces The...
  21. C

    Importance of differential geometry in physics?

    How important is differential geometry in physics? Can someone give me some applicable fields?
  22. S

    Differential geometry vs differential topology

    in a nutshell, what is the difference between those two fields?
  23. A

    Differential Geometry Relations, relating to plasma physics.

    The context of this question is looking at straggling in plasma. I was told there was a simple differential geometry relationship between the following entities: dE/dx, dE/dy and dE/dt, where x,y are distance in perpendicular directions (axes on a plane), t is time and I'm using E to...
  24. P

    I want to study algebraic geometry and differential geometry

    I want to study algebraic geometry and differential geometry, what should I learn beforehand what is the relation between abstract algebra and homological algebra:confused:
  25. S

    How to learn differential geometry?

    Hello everyone! I just wanted to ask a question about how I should study for differential geometry. Now, as I have it, I've got a few suggestions for books, of which two stand out prominently: 1. John Lee's Introduction to smooth manifolds 2. De Carmo Which one would be best for self study...
  26. J

    I with differential geometry computing connection forms. Please respond

    I need help with Part (b). I finished part (a) and attached it as well. My issue comes from how to apply the definition of connection forms to compute them. The definition states: Let E_1, E_2, E_3 be a frame field on R^3. For each tangent vector v at R^3 at the point p let \omega_{ij}(v )=...
  27. W

    Books on differential geometry

    As the title says. Can anyone recommend me some good books for differential geometry(preferably ones with proofs and examples/exercises)?
  28. Z

    Cartan's 1924 mystery formula differential Geometry

    hey all, in this book ; https://www.amazon.com/Geometry-Riemannian-Spaces-Lie-Groups/dp/0915692341/ref=sr_1_1?ie=UTF8&qid=1348590926&sr=8-1&keywords=geometry+of+riemannian+++spaces++cartan On page 178 ( which I attach a snapshot of it) Cartan had introduced a formula (see in the snapshot...
  29. CJ2116

    Topology and Differential Geometry texts for General Relativity

    Hi everyone, I was wondering if I could some advice from anyone who has some experience with higher level general relativity. Any help would be greatly appreciated! Some background: I'm currently working through Robert Wald's General Relativity and am struggling a lot with the "advanced...
  30. J

    A problem in Elementary Differential Geometry

    My teacher has defined U_1 = \langle1, 0, 0\rangle, U_2 = \langle0, 1, 0\rangle, and U_3 = \langle0, 0, 1\rangle. So it seems like the function maps L(\langle1, 0, 0\rangle, \langle0, 1, 0\rangle) = a, L(\langle1, 0, 0\rangle, \langle0, 0, 1\rangle) = b,, and L(\langle0, 1, 0\rangle...
  31. A

    Will a mathematician understand GR if he knows differential geometry well?

    The title says everything. Can a mathematician do fruitful research in general relativity if he masters differential geometry and manifolds?
  32. T

    Diff Geom in Complex Spaces: Hermitian, Anti-Symmetric & Affine Connections

    Are there any good papers or books that go over our current understanding of differential geometry for 2-dimensional complex spaces? Hermitian vs anti-symmetric metric tensors, dealing with complex conjugates, and defining affine connections? Yes, I've already hit up Google, so I was hoping...
  33. B

    Looking to Prepare for Metric Differential Geometry

    This is the course description: I want to take this class because the professor comes highly recommended, but I'm a little worried that I won't be entirely prepared for it. Normally this class requires Real Analysis as a prerequisite, and even though the professor explicitly states that...
  34. S

    Books to Prepare for Differential Geometry

    What books should I read as prerequisites for Spivak's Differential Geometry Series? Trying to pick up Diff Geom for graduate physics but right now it is pretty daunting. I've got single variable calculus and linear algebra under my belt. Just looking for bare minimum requirements here to...
  35. C

    Complex Analysis or Differential Geometry first

    I have to choose which math course I'm going to take next term. I want to take both but I'm already taking two physics courses and my college's distribution requirements require that I take an English next term... bleh... I could audit one of the physics and then take both math courses, but that...
  36. I

    Importance of Lie algebra in differential geometry

    Hi all, I've been wondering about this for some time. While I am only familiar with the basics of differential geometry, I have come across the Lie bracket commutator in a few places. Firstly, what is the intuitive explanation of the Lie bracket [X,Y] of two vectors, if there is one? In...
  37. M

    Coordinate based vs non-coordinate based differential geometry

    Hello Everyone, I am just wondering what the difference in these is. Could someone please give a brief example of non-coordinate based differential geometry vs the equivalent in coordinate based, or explain the difference (whichever is easier)? Also, what advantages does one have over the...
  38. R

    Prereq for Differential Geometry

    I am an Astrophysics undergrad, and will be taking Classical Differential Geometry I & II. Are there any classes that will make understanding Differential Geometry easier. I can chose from: -Introduction To Abstract Algebra -Introduction To Mathematical Analysis -Introduction To Real...
  39. H

    What are the contents of the Differential Geometry Library?

    http://digi-area.com/DifferentialGeometryLibrary/ includes over 580 objects for differential geometry and its applications. Moreover here is 380 Exact Solutions of Einstein's Field Equations. The formulas are represented in different forms: metric from, Contravariant Newman–Penrose Tetrad...
  40. J

    Very elementary differential geometry questions

    I have decided to attempt to pick up some differential geometry on my own, and I am trying to get some traction on the subject which I do by trying to reduce it to familiar and simple cases. This is a trivial case, I know, but it will go a long way in advancing my understanding. Suppose the...
  41. S

    Differential Geometry, curve length

    Homework Statement Homework Equations L[c]:=\int_{a}^{b}(\sum_{i,j=1}^{2}g_{ij}(c(t))c_{i}'(t)c_{j}'(t))^{1/2}dt The Attempt at a Solution So g_{ij}(x,y)=0 for i{\neq}j, c_{1}'(t)=-Rsin(t), c_{2}'(t)=Rcos(t) so...
  42. P

    Introductory Differential Geometry Book With Lots of Intuition

    So I took an analysis class which covered chapters 9 and 10 of Rudin's PMA, for those of you who don't know that's multivariable analysis and differential forms, and I have taken a course in vector calculus but never a proper course on differential geometry. Thus my introduction to the subject...
  43. Z

    Lecture notes in The Differential geometry of Gauge theory?

    Hi all , How can I find lecture notes on ArXiv ? I was looking for lecture notes on Yang-mills theories treated in the language of differential geometry but didn't succeed till now . Can some one recommend me some good resource for it?
  44. M

    Differential Geometry Surface with planar geodesics is always a sphere or plane

    Homework Statement Show that if M is a surface such that every geodesic is a plane curve, then M is a part of a plane or a sphere. Homework Equations - If a geodesic, \alpha, on M is contained in a plane, then \alpha is also a line of curvature. - Let p be any point on a surface M and...
  45. S

    Differential Geometry, easy question, weird hint making me doubt myself

    Homework Statement Homework Equations From my notes: (\psi_{*}v)_{k}(x)=\sum_{i=1}^{n}v_{i}(x)\frac{{{{{\partial}}}}{\psi_{k}(x)}}{{x_{i}}} The Attempt at a Solution Okay so i) is fine (ignoring the typo in the question) but I'm a bit confused about ii) I don't see any need...
  46. B

    Differential Geometry: Lie derivative of tensor fields.

    Homework Statement Let M be a differentiable manifold. Let X and Y be two vector fields on M, and let t be a tensor field on M. Prove \mathcal{L}_{[X,Y]}t = \mathcal{L}_X\mathcal{L}_Yt -\mathcal{L}_Y\mathcal{L}_Xt Homework Equations All is fair game, though presumably a coordinate-free...
  47. U

    What exactly is differential geometry?

    Hello. I'm new here and I'm not sure if I should post this topic here or in general math, or anywhere else. Feel free to move the topic elsewhere if needed. Just a bit of explanation first. I'm not from USA (so forgive any grammar errors) and I don't understand completely your academic...
  48. I

    Usefulness of Differential Geometry?

    My university is offering a Differential Geometry course next semester and while I am interested in the subject, I do not plan to take the class unless it has practical use for me. I have no interest in doing theoretical work. Does differential geometry serve any use to applied physics/engineering?
  49. 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)
  50. 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...
Back
Top