By Chuan-Chih Hsiung
The origins of differential geometry return to the early days of the differential calculus, whilst one of many primary difficulties was once the decision of the tangent to a curve. With the improvement of the calculus, extra geometric purposes have been acquired. The relevant 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 leap forward used to be taken by means of Karl FriedrichGauss (1777-1855) along with his improvement of the intrinsic geometryon a floor. this concept of Gauss used to be generalized to n( > 3)-dimensional spaceby Bernhard Riemann (1826- 1866), hence giving upward push to the geometry that bears his identify. This booklet is designed to introduce differential geometry to starting graduate scholars in addition to complex undergraduate scholars (this advent within the latter case is necessary for remedying the weak point of geometry within the traditional undergraduate curriculum). within the final couple of many years differential geometry, besides different branches of arithmetic, has been hugely built. during this booklet we are going to learn in basic terms the normal subject matters, particularly, curves and surfaces in a 3-dimensional Euclidean house E3. in contrast to so much classical books at the topic, besides the fact that, extra realization is paid the following to the relationships among neighborhood and international homes, as against neighborhood homes in basic terms. even if we limit our cognizance to curves and surfaces in E3, so much international theorems for curves and surfaces in this e-book will be prolonged to both greater dimensional areas or extra common curves and surfaces or either. in addition, geometric interpretations are given besides analytic expressions. this may permit scholars to utilize geometric instinct, that's a important software for learning geometry and similar difficulties; this type of software is seldom encountered in different branches of arithmetic.
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Extra resources for A first course in differential geometry
The maximal ﬂat Radon transform for functions on X assigns to a real-valued function f on X the function fˆ on Ξ, whose value at a torus 42 II. RADON TRANSFORMS ON SYMMETRIC SPACES Z ∈ Ξ is the integral fˆ(Z) = f dZ Z of f over Z. Clearly this transform is injective if every function on X satisfying the Guillemin condition vanishes. The X-ray transform for functions on X assigns to a real-valued function f on X the function fˇ on Ξ , whose value at a closed geodesic γ ∈ Ξ is the integral fˇ(γ) = f.
The Guillemin and zero-energy conditions Let (X, g) be a Riemannian manifold. For p ≥ 0, we consider the symmetrized covariant derivative p : S p T ∗ → S p+1 T ∗ , Dp = DX which is the ﬁrst-order diﬀerential operator deﬁned by 1 (D u)(ξ1 , . . , ξp+1 ) = p+1 p+1 (∇u)(ξj , ξ1 , . . , ξˆj , . . , ξp+1 ), p j=1 for u ∈ S p T ∗ and ξ1 , . . , ξp+1 ∈ T . The operator D0 is equal to the exterior diﬀerential operator d on functions, and the operator D1 was already introduced in §1, Chapter I. 3) X for all f ∈ C ∞ (X) and all parallel vector ﬁelds ζ on X; therefore, if u is a symmetric p-form on X, we have (Dp u)(ζ1 , .
12. We say that a compact locally symmetric space X is rigid in the sense of Guillemin (resp. inﬁnitesimally rigid) if the only symmetric 2-forms on X satisfying the Guillemin (resp. the zero-energy) condition are the Lie derivatives of the metric g. If X is a compact locally symmetric space X and p ≥ 0 is an integer, we consider the space Zp of all sections of C ∞ (S p T ∗ ) satisfying the zeroenergy condition. 6, we have the inclusion Dp C ∞ (S p T ∗ ) ⊂ Zp+1 . 4), we see that the inﬁnitesimal rigidity of the compact locally symmetric space X is equivalent to the equality D1 C ∞ (T ∗ ) = Z2 .
A first course in differential geometry by Chuan-Chih Hsiung