By Katsuhiro Shiohama, Takashi Shioya, Minoru Tanaka

This self sustaining account of recent rules in differential geometry indicates how they are often used to appreciate and expand classical leads to indispensable geometry. The authors discover the effect of overall curvature at the metric constitution of whole, non-compact Riemannian 2-manifolds, even though their paintings should be prolonged to extra normal areas. every one bankruptcy positive factors open difficulties, making the amount an appropriate studying relief for graduate scholars and non-specialists who search an creation to this contemporary region of differential geometry.

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**The Geometry of Total Curvature on Complete Open Surfaces**

This self sustaining account of recent principles in differential geometry indicates how they are often used to appreciate and expand classical ends up in critical geometry. The authors discover the impression of overall curvature at the metric constitution of whole, non-compact Riemannian 2-manifolds, even though their paintings should be prolonged to extra basic areas.

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**Extra info for The Geometry of Total Curvature on Complete Open Surfaces**

**Example text**

Tk = so that Ji := J |[ti−1 ,ti ] for all i = 1, . . , k is smooth. Let f i : [ti−1 , ti ] → [0, 1] be a smooth function such that f i (ti−1 ) = f i (ti ) = 0 and f i > 0 on (ti−1 , ti ), and set Z i := f i (Ji + R(Ji , γ˙ )γ˙ ). Then setting Z (t) := Z i (t) for t ∈ [ti−1 , ti ] we see that k ti i=1 ti−1 0 = I (J, Z ) = f i Ji + R(Ji , γ˙ )γ˙ 2 dt. Thus Ji for each i = 1, . . , k − 1 is a Jacobi ﬁeld along γ |[ti−1 ,ti ] . To prove the smoothness of J we choose a vector ﬁeld Z ∈ X⊥ such that for each i = 1, .

By eliminating all the compact components of ∂ M, we can construct from M a new surface M1 such that M1 coincides with M outside a compact set. Namely, ∂ M1 has k − components each of which is a divergent curve and M1 has tubes. Such an M1 is obtained by pasting a disk domain on-to each component of ∂ M0 . Let g1 be a complete Riemannian metric on M1 such that g = g1 on M \ Ci for some i ≥ 1. 12) where χ (∂ M1 ) = χ(∂ M) is the number of (unbounded) components of ∂ M1 . We see from this fact that we need only to discuss the noncompact components of ∂ M.

1) t 2 + 1)2 Setting tan ϕ := k for ϕ ∈ (0, π/2), we see that ν(M) omits (π/2 − ϕ)-balls around the north and south poles of S2 (1) and hence that c(M) = −4π sin ϕ. Again, using direct evaluation, c(M) = lim 2π t t→∞ 0 = −4π √ −t k k2 + 1 −k 2 k 2 (k 2 + 1)t 2 + 1 3/2 dt dθ = −4π sin ϕ. 2 The classical theorems of Cohn-Vossen and Huber The classical well-known theorem due to Cohn-Vossen will be stated under more general assumptions. We show here that the Huber theorem is derived directly from the Cohn–Vossen theorem.