By Yu. Aminov

It is a finished presentation of the geometry of submanifolds that expands on classical leads to the speculation of curves and surfaces. The geometry of submanifolds starts off from the assumption of the extrinsic geometry of a floor, and the speculation reports the location and houses of a submanifold in ambient area in either neighborhood and international features. Discussions comprise submanifolds in Euclidean states and Riemannian house, minimum submanifolds, Grassman mappings, multi-dimensional commonplace polyhedra, and isometric immersions of Lobachevski area into Euclidean areas. This quantity additionally highlights the contributions made by way of nice geometers to the geometry of submanifolds and its parts of program.

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**Extra resources for The Geometry of Submanifolds**

**Example text**

X") the coefficients of the second fundamental form are L; = z,,,,/ l + (grad z12. 3) 2 - (x4) 2 the asymptotic lines satisfy the equation (dx') 2 + (dr 2) 22 - (d%3) 2 - (dr4) 2 = 0. The p-parametric curve x1 =a cos a, gyp, x2 = a sin a, cp, a; = b COs a2cp, x4 = b sin a2rp, with a. b, a,, a2 constants, satisfy the equation above when a condition (aa,) 2 = (bat) 2 is satisfied. These curves are closed when a, /a2 is a rational number. The curve considered in the example lies in a sphere of radius v/a2 -+b2.

Let xi, y,, zi (i = 1, 2) be Cartesian coordinates of the endpoints. The location of the segment can be described by six coordinates x1, y1, 21,x2, }'2, z2. They fix the point in E6. Subject this coordinates to conditions: ''r(xny,, 21) = 0, i=1,2; 4'3 = 2 [(xi - x2)2 + U'1 -1'2)2 + (21 - 22)2 - a2]= 0. GENERAL PROPERTIES OF SUBMANIFOLDS 25 IIGURE 5 The first and second conditions are the equations of the surfaces to which the endpoints of the segments belong. So, we have an implicit representation of some submanifold in E6.

By means of the Weingarten decomposition, do = n,,, du' _ -L; dit'gikr,rk. For the principal direction the following holds: (L;,,-Agy)du'=0. Hence do = -Agydug/kr". = -Abler,,. du' = -A dr. Thus, Rodrigues' formula holds -- the differential of a unit normal along the principal direction is parallel to this direction: do = -A dr. l)2)4' where (') means differentiating with respect to t. Prove that coordinate curves are lines of curvature. 4 Integrability Condition for Principal Directions In contrast to the two-dimensional case, many-dimensional hypersurfaces do not, in general, possess coordinates that the coordinate hypersurface would be orthogonal to the principal directions.