Download A first course in differential geometry by Chuan-Chih Hsiung PDF

By Chuan-Chih Hsiung

The origins of differential geometry return to the early days of the differential calculus, while one of many primary difficulties used to be the selection of the tangent to a curve. With the improvement of the calculus, extra geometric purposes have been received. The primary members during this early interval have been Leonhard Euler (1707- 1783), GaspardMonge(1746-1818), Joseph Louis Lagrange (1736-1813), and AugustinCauchy (1789-1857). A decisive breakthrough used to be taken via Karl FriedrichGauss (1777-1855) along with his improvement of the intrinsic geometryon a floor. this concept of Gauss was once generalized to n( > 3)-dimensional spaceby Bernhard Riemann (1826- 1866), hence giving upward thrust to the geometry that bears his identify. This booklet is designed to introduce differential geometry to starting graduate scholars in addition to complicated undergraduate scholars (this creation within the latter case is critical for remedying the weak spot of geometry within the ordinary undergraduate curriculum). within the final couple of many years differential geometry, besides different branches of arithmetic, has been hugely constructed. during this e-book we'll research merely the normal issues, specifically, curves and surfaces in a 3-dimensional Euclidean area E3. not like such a lot classical books at the topic, even though, extra awareness is paid right here to the relationships among neighborhood and worldwide houses, as against neighborhood houses in simple terms. even supposing we limit our consciousness to curves and surfaces in E3, such a lot international theorems for curves and surfaces in this booklet could be prolonged to both better dimensional areas or extra normal curves and surfaces or either. furthermore, geometric interpretations are given besides analytic expressions. this can let scholars to utilize geometric instinct, that's a necessary device for learning geometry and comparable difficulties; this type of instrument is seldom encountered in different branches of arithmetic.

Show description

Read Online or Download A first course in differential geometry PDF

Similar differential geometry books

Foundations of mechanics

Within the Spring of 1966, I gave a chain of lectures within the Princeton collage division of Physics, geared toward fresh mathematical ends up in mechanics, particularly the paintings of Kolmogorov, Arnold, and Moser and its software to Laplace's query of balance of the sun procedure. Mr. Marsden's notes of the lectures, with a few revision and enlargement by way of either one of us, grew to become this ebook.

Lectures on classical differential geometry

First-class short advent provides primary conception of curves and surfaces and applies them to a few examples. subject matters contain curves, thought of surfaces, primary equations, geometry on a floor, envelopes, conformal mapping, minimum surfaces, extra. Well-illustrated, with considerable difficulties and ideas.

New Developments in Differential Geometry (Mathematics and Its Applications)

This quantity includes thirty-six learn articles provided at the Colloquium on Differential Geometry, which used to be held in Debrecen, Hungary, July 26-30, 1994. The convention used to be a continuation in the sequence of the Colloquia of the J? nos Bolyai Society. the diversity lined displays present task in differential geometry.

Riemannian geometry during the second half of the twentieth century

In the course of its first hundred years, Riemannian geometry loved regular, yet undistinguished development as a box of arithmetic. within the final fifty years of the 20th century, in spite of the fact that, it has exploded with task. Berger marks the beginning of this era with Rauch's pioneering paper of 1951, which incorporates the 1st actual pinching theorem and an grand bounce within the intensity of the relationship among geometry and topology.

Additional resources for A first course in differential geometry

Example text

15) then yield E E h = (π∗ E) − A∗E h c˙v − A∗c˙h E v + σ(c˙v , E v ), v = Ac˙h E h − SE h c˙v + E v v . 16) provides another proof of the fact that F splits locally if A and S are identically zero. One can argue this using either horizontal or vertical fields: for example, if E is a parallel field along c with E(t0 ) vertical, then E v is parallel, so E = E v is always vertical. Thus, V is invariant under parallel translation, and the claim follows from de Rham’s holonomy theorem. 16) applied to the tangent field c˙ of a curve c in M yields ˙ − 2A∗c˙h c˙v + σ(c˙v , c˙v ), c˙ h = (π∗ c) c˙ v = −Sc˙h c˙v + c˙v v .

Next, set V1 (t) := {J ∈ V | J(t) = 0}, V2 (t) := {J (t) | J ∈ V, J(t) = 0}. The map V1 (t) → V2 (t) which sends J ∈ V1 (t) to J (t) is linear, surjective by definition, and has zero kernel. Thus V1 (t) ∼ = V2 (t). For any fixed t0 ∈ R, let J1 , . . , Jk be a basis of V1 (t0 ), and extend it to a basis J1 , . . , Jl of V, where l := dim V. Then Jk+1 (t0 ), . . , Jl (t0 ) is a basis of {J(t0 ) | J ∈ V}, and dim V = l = k + l − k = dim V1 (t0 ) + dim{J(t0 ) | J ∈ V} = dim V2 (t0 ) + dim{J(t0 ) | J ∈ V} = dim V (t0 ).

Jl of V, where l := dim V. Then Jk+1 (t0 ), . . , Jl (t0 ) is a basis of {J(t0 ) | J ∈ V}, and dim V = l = k + l − k = dim V1 (t0 ) + dim{J(t0 ) | J ∈ V} = dim V2 (t0 ) + dim{J(t0 ) | J ∈ V} = dim V (t0 ). To prove the second statement, let t0 and J1 , . . , Jl be as above. We will show that there is some > 0 such that J1 (t), . . , Jl (t) are linearly independent for each t ∈ (t0 , t0 + ). By assumption, Jk+1 (t0 ), . . , Jl (t0 ) are linearly independent, and thus remain so for t close enough to t0 .

Download PDF sample

Rated 4.55 of 5 – based on 26 votes