# Questions tagged [differential-geometry]

For questions about the discipline that uses differential calculus and linear algebra to study geometrical problems.

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### What did Jacobi, who lived before Riemann, have to do with the equation and theorem named after him in Riemannian geometry?

In Riemannian geometry we have two very important things named after Jacobi: the Jacobi equation $J''=R(\gamma',J)\gamma'$ and Jacobi's theorem which states geodesics never minimize past conjugate ...
499 views

### Where can I find the original presentation of the proof, due to Grothendieck, of the $\bar\partial$-Poincaré lemma?

In complex geometry, there is the a lemma, analogous to the Poincaré lemma in (real) differential geometry, which states that a $(p,q)$-form that is $\bar\partial$-closed is locally $\bar\partial$-...
3k views

### How was Einstein led to make a contact with Differential Geometry for his theory of General Relativity?

General Relativity was developed with Differential Geometry as the tool. How was Einstein led to make a contact with Differential Geometry for his theory of General Relativity? Who suggested him to ...
237 views

### why was the hairy ball theorem important

In Topology courses one learns An even dimensional sphere does not possess any continuous field of unit vectors What is the importance of this result? I can't think of any applications off the top ...
111 views

### on the classification of singular points

After reading this question and the answers to it, I am interested o know who were the first mathematicians who started classifying singular points of curves: i.e. different kind of nodes, of cusps ...
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### History of the derivative/tangent of a curve

I just want to know the history of the derivative. Whenever I Google for it, I find the history of calculus or the tangent of a curve. However, they barely touch upon what happened before Leibniz and ...
87 views

### Whence originates the use of the nabla (∇) for a connection or covariant derivative?

Who introduced it, when, where, and with what if any rationale? (Note that I am not asking about the origin of the nabla symbol, which is covered here.)
384 views

### How did the exterior product get its symbol?

As per the title: where did the notation $a\wedge b$ for the exterior product of $a$ and $b$ originate, and/or who popularised it? I'm especially interested in motivation for the choice of this symbol ...
1k views

### How did the exponential map of Riemannian geometry get its name?

I've read in several books, including Milnor's Morse Theory and Petersen's Riemannian Geometry, that the exponential map in Riemannian geometry is named so because it agrees with the exponential map ...
The space of sections of a bundle $\pi: E \to B$ is commonly denoted $\Gamma(E)$. (Note that the graph of a function $f$ is $\Gamma(f):=\left\{(x,f(x))\right\}$, and a particular section $\sigma: B \... 5 votes 1 answer 837 views ### What is the origin of the use of "g" for a Riemannian metric? I am asking about the reason for the use of this letter, if known, as well as the initial occasion of its use. Ideas that have been suggested concerning the former include: That it stands for ... 0 votes 1 answer 332 views ### The debauch of indices: translation request Finally I found the source of the dictum "debauches of indices". It is most often used in singular ""debauch", as in Spivaks's Vol.II p.211. The original is from the first ... 4 votes 1 answer 160 views ### Did anybody consider the product of the principal curvatures before Gauss? Gauss proved that the so-called Gaussian curvature is an intrinsic invariant of the surface, even though it is defined extrinsically as the product of the principal curvatures of a Euclidean embedding.... 10 votes 1 answer 1k views ### How was curvature originally defined and calculated? I am interested in the early history of curvature. Who defined it first and when, who came up with the name, how was it calculated before mathematicians used calculus to define$k=|α''(s)|$? Are there ... 5 votes 0 answers 266 views ### Is$\Gamma^i_{jk}$the Christoffel symbol or the Christoffel symbols? For years, I have been perplexed that the expression$\Gamma^i_{jk}$is often referred to in the plural as "the Christoffel symbols", although sometimes it is referred to in the singular as "the ... 9 votes 4 answers 891 views ### Was the term "manifold" (or its German equivalent) chosen with the verb "to fold" in mind? Recently I came across several papers of Monge and Lagrange, around the end of the 18th century, considering developable surface as 'folded' planes, using specifically the word "plié" (i.e. folded). ... 6 votes 2 answers 487 views ### Did Gauss formulate, or at least know of, the full essence of the Gauss-Bonnet Theorem? I know that a special case of the Bonnet theorem, called the Theorema Elegantissimum, was proved by Gauss in his 1827 treatise on differential geometry. This was a theorem that dealt with the ... 8 votes 3 answers 594 views ### Where did Cartan introduce his notation for basis vectors and covectors? There is a notation used in differential geometry and general relativity in which the partial derivative operators$\partial_\mu\$ are used as the basis for the space of contravariant vectors, and ... 