Geodesic
On Gaussian and Geodesic Curvature of Riemannian Manifolds
Canadian journal of mathematics, Tome 26 (1974) no. 3, pp. 629-635

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In [1], S. S. Chern gave a very elegant and simple proof of the Gauss-Bonnet formula for closed (i.e. compact without boundary) oriented Riemannian manifolds of even dimension: Here, c is a suitable constant depending on the dimension of M and Ω is an n-form (n = dim M) which may be calculated from its curvature tensor. W. Greub gave a coordinate-free description of this integrand Ω (cf. [4]). 
Rummler, Hansklaus. On Gaussian and Geodesic Curvature of Riemannian Manifolds. Canadian journal of mathematics, Tome 26 (1974) no. 3, pp. 629-635. doi: 10.4153/CJM-1974-060-x
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[1] 1. Chern, S. S., A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. 45 (1944), 747–752. Google Scholar

[2] 2. Chern, S. S., On the curvatura intégra in a Riemann manifold, Ann. of Math. 46 (1945), 674–684. Google Scholar

[3] 3. Chern, S. S., On curvature and characteristic classes of a Riemann manifold, Abh. Math. Sem. Univ. Hamburgh (1955), 117-126. Google Scholar

[4] 4. Greub, W. H., Zür Théorie der linear en Uebertragungen, Ann. Acad. Sci. Fenn. Ser. A. I. 846 (1964), 3–32. Google Scholar

[5] 5. Greub, W. H., Multilinear algebra (Springer, Berlin, Heidelberg, New York, 1967), Google Scholar

[6] 6. Hicks, N. J., Notes on differential geometry (Van Nostrand, New York, 1965). Google Scholar

[7] 7. Holmann, H. and Rummler, H., Differential﹜or men, Bibliographisches Institut, Mannheim, 1972. Google Scholar

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