CONTENTS:- 1. SOME PRELIMINARIES, 2. TENSOR ALGEBRA, 3. RIEMANNIAN METRIC, 4. CHRISTOFFEL SYMBOLOS COVARIANT DIFFERENTIATION, 5. THE FOUNDATIONS, 6. CURVES IN SPACE, 7. CONCEPT A SUREFACE AND ENVELOPES AND DEVELOPMENT, 8. FUNDAMENTAL FORMS AND CURVATURE OF SURFACES, 9. LOCAL NON - INTEINSIC PROPERTIES OF A SURFACE, 10. CON. ASYMPTOTIC LINES FUNDAMENTAL EQU. OF SURFAVEC THEORY AND PARALLEL SURFACES, 11. GAUSS EQUATION, GAUSS COEFFICIENTS AND GEODESICS, 12. TENSOR ANALYSIS