Basic understanding of differential geometry

[shounakbhatta]

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?
(3) Is it that as General relativity deals with curved space time, hence differential geometry comes into the scene?
(4) Does it include differential calculus and 3-d geometry?

Thanks

 

[quasar987]

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?
(3) Is it that as General relativity deals with curved space time, hence differential geometry comes into the scene?
(4) Does it include differential calculus and 3-d geometry?

Thanks

(1) Yes, it is exactly that!
(2) The so-called "riemannian geometry" unifies and generalizes the "classical geometries" (euclidean, spherical and hyperbolic geometries), in that their realization all arise as special cases. Another difference is that in classical geometry, one studies a set of axioms. In riemannian geometry, one studies a manifold with a riemannian metric on it.
(3) Yes, more precisely, Einstein was looking for a "coordinate independent" way to formulate the laws of physics (according to the postulate of general relativity in which he had faith). With the help of his mathematician friend (Maurice or Michel something?), he discovered that the tensor calculus on manifolds was perfectly suited to the task.
(4) Yes, the basic idea behind differential geometry is to use differential calculus to study curves and surfaces and manifolds.

 

[shounakbhatta]

Hello,

Now let me refine few of my queries. Euclidean geometry -> Analytic geometry ->Non-euclidean geometry --> Differential geometry. The flow goes something like this, right?

Secondly, to get a invariant geometry (can I call a co-ordinate independent geometry in that way?), differential geometry is used?

So, if I plot a triangle on a sphere, the straight lines becomes curved. So to get lengths, etc.we need to differentiate them, hence differential geometry. Is that so?

I don't have any knowledge in topology, just plain calculus. Can I learn diff.geometry?

What are the per-requisites to learn diff.geometry?

Thanks

 

[micromass]

Hello,

Now let me refine few of my queries. Euclidean geometry -> Analytic geometry ->Non-euclidean geometry --> Differential geometry. The flow goes something like this, right?

I would say that non-euclidean geometry is a part of differential geometry although it depends.

Secondly, to get a invariant geometry (can I call a co-ordinate independent geometry in that way?), differential geometry is used?

You can perfectly practice differential geometry in a very coordinate dependent way. This is what is done classically, but it's not very pretty. The coordinate independent way is quite elegant, but also more abstract.

So, if I plot a triangle on a sphere, the straight lines becomes curved. So to get lengths, etc.we need to differentiate them, hence differential geometry. Is that so?

Those lines on the sphere are called geodesics. They are studied in differential geometry.

I don't have any knowledge in topology, just plain calculus. Can I learn diff.geometry?

Yes, although I also recommend knowing some linear algebra. I recommend the books by Do Carmo and O'Neill.
https://www.amazon.com/dp/0132125897/?tag=pfamazon01-20
https://www.amazon.com/dp/0120887355/?tag=pfamazon01-20

What are the per-requisites to learn diff.geometry?

Classical differential geometry just requires a good knowledge of calculus and linear algebra. This will cover curves and surfaces, but only a little bit of manifolds. When you studied that, you basically start with a study of manifold theory and modern differential geometry. For this, you need topology and a bit of real analysis. I highly recommend the books by John Lee. His first book covers topology quite well: https://www.amazon.com/dp/1441979395/?tag=pfamazon01-20
His second and third books cover smooth manifolds and Riemannian manifolds.

 

[yenchin]

(
(3) Yes, more precisely, Einstein was looking for a "coordinate independent" way to formulate the laws of physics (according to the postulate of general relativity in which he had faith). With the help of his mathematician friend (Maurice or Michel something?), he discovered that the tensor calculus on manifolds was perfectly suited to the task.
.

Marcel Grossmann

 

[shounakbhatta]

Hello Micromass,

You have written that non-Euclidean geometry is also a part of diff.geometry, although it depends. Can you please explain me on that?

Thanks.

 

[WannabeNewton]

Hello Micromass,

You have written that non-Euclidean geometry is also a part of diff.geometry, although it depends. Can you please explain me on that?

Thanks.

It depends on how you define non-Euclidean geometry (what you consider to be the content of the subject). For sure the study of curved manifolds, where euclid's parallel postulate doesn't hold in general, counts as non-Euclidean geometry so differential geometry is arguably a part of non-euclidean geometry.

 

[lavinia]

Hello Micromass,

You have written that non-Euclidean geometry is also a part of diff.geometry, although it depends. Can you please explain me on that?

Thanks.

Historically Euclidean geometry was defined axiomatically. This first happened in Ancient times, long before calculus was discovered.

Non euclidean geometry rose in the 18'th century - I think - from questions about why the parallel postulate, the postulate that says that there is only one parallel to a line through a given point in a plane - remained unproved after two thousand years of failed attempts. These researches did not use calculus and I suspect that the first model of a non-Euclidean plane geometry was axiomatic. In any case, non-Euclidean plane geometry can be defined without calculus, and without the Differential Geometric notions of geodesics and curvature.

During the 18'th century differential geometry thrived as the study of curved surfaces in space but its connection to non-Euclidean geometry was unknown until someone discovered that non-Euclidean geometry could be modeled as a space of constant negative curvature.

For most of history, Europeans believed that the geometry of space was intrinsic to the idea of space itself so that all geometry could be derived from simple notions of space. This is probably why geometry was an axiomatic system. When non-Euclidean geometry was discovered it was realized that there were two possible plane geometries and researchers such as Gauss tried to measure large triangles on the Earth to determine which one is correct. But still the considerations were about the axioms of space.

Later probably with Riemann, a new idea came up, that of a manifold without a given geometry but which could be given many geometries. This new idea rejected the assumption that the geometry of space is intrinsic. This new idea led to modern differential geometry and largely displaced the interest in the axiomatic approach.